Groups with Prescribed Quotient Groups and Associated Module Theory (Series in Algebra)

= y{giy{

+ ... + gmym)

= y{{g\S)j\

+ ... + (gm0)ym)

= yy~l = 1.

Hence l(p<5) = yd = I and p8 becomes an FG - isomorphism of D9 on De-

The Structure of Simple Modules over Abelian Groups

21

Therefore, without loss of generality we can assume that l p = 1. Given x e Dv and g e G, we have (x(g)p = g9 for all g s G, what means 0 = (pp. Furthermore, we will prove that p preserves the multiplication. It is sufficient to prove this for a pair of elements x, z E G
(xp)(zp).

Consequently p is an isomorphism between the subfields Cv and Cs- The converse statement is obvious. Note that if F is a locally finite field, then G

= GO. It follows that CO = C(p. In other words, p e Gal(Ce/F). 2.4. Corollary i e i G i e a« abelian group of finite 0 - rawft, D a Dedekind domain with infinite Spec(D). (1) If A is a simple DG - module, then Ann D(A) = P e Spec(D) and t(G/Cc{A)) is a locally cyclic pi- group where p = char{DIP). Moreover, if DIP is a locally finite field, then GICQ{A) is aperiodic group. (2) Conversely, if P e Spec(D) and DIP is a locally finite field, H is a subgroup ofG such that GIH is a locally cyclic p> -group where p = char{DIP), then there exists a simple DG - module A with the following properties: CG{A) = H,AnnD(A) = P. (3) IfP e Spec(D) and DIP is not locally finite, H is a subgroup ofG such that t(G/H) is a locally cyclic pi- group where p = char(D/P), then there exists a simple DG - module A with the following properties: CQ{A) = H, AnnoiA) = P. Proof The statement (1) implies from Corollary 1.16, Corollary 2.2, and Corollary 1.21. Conversely, let P e Spec{D) and DIP is a locally finite field, K an algebraic closure of F = DIP. Then U{K) is a divisible locally cyclic pi- group. Then there is a homomorphism 6 : G —> U(K) such that KerO = H. By Theorem 2.3 there exists a simple FG - module A with the property CG{A) = H. We can consider A as a DG - module such that Anno(A) = P. Let DIP be non-(locally finite). Then U(K) is a divisible abelian group and r0(U(K)) = \F] (see, for example, [KG, Chapter 4, Theorem 6.1]). The subgroup H satisfying the current hypothesis is the kernel of the homomorphism 0 : G —• U(K) and we can iterate the same arguments given above.

Simple Modules

22

2.5. Corollary Let D be a Dedekind domain with finite Spec(D), G an abelian group of finite 0 - rank. (1) If A is a simple DG- module, then t(GICG(A)) is a locally cyclic pi-group where p = char{DIAnnD(A)). (2) Consequently, let I be a prime ideal of D {we allow I to be < 0 >), H a subgroup ofG such that t(G/H) is a locally cyclicp1- group where p = char{D/I). Then there exists a simple DG - module A with the properties: AnnD{A) = I,CG(A) = H. Proof Since (1) follows from Corollary 2.2, it suffices to prove (2). If / e Spec(D), then F = DII is a field and we can take an algebraic closure K of F. Just as in Corollary 2.4 there is a homomorphism 6 : G —• U(K) such that Kerd = H. Thus, theorem 2.3 yields that there exists a simple FG -module A such that CG(A) = H. We can consider A as a DG - module with this meaning Anno(A) = /. Therefore, we only have to consider the case / = < 0 >. Choose in GIH a maximal Z-independent set {x\H,... ,xrH} and put TIH = t{GIH), Y = < T,xi,... ,Xr >, Y\ =< T,X\,... ,Xr-\ >.

Then Y = Y\ x< xr >. Since Spec{D) is finite, D is a principal ideal domain (see, for example, [ZS, Chapter V, Theorem 16]). There is an ideal L of group ring D < xr > such that (D < xr >)IL = F\ where F\ is the field of fractions of D and C<xr>{D < xr > IV) = < 1 > (see Example 1.22). As in the other cases, we take an algebraic closure K of Fi and the homomorphism 6 : Y\ —* U(X) with the property KerO = H. Then Ai = F\[(Yi)0] is a subfield of K, thus .4i4s a simple F\Y\ - module with the property Cy^Ai) = H. If a e A\, then a = d\(c\d) + ... + d„(c„6) where dj e F\,d 6 Yul < i< n. Put axr = {dixr){cxff) + ... + (d„xr)(c„0). Then A\ becomes a module over the group Y\ x < xr > = Y, moreover, Aj is a simple DY- module and Cy(A\) = H. Let B - A i @DY DG, A a DG - composition factor of B. Since B = 0 S(A i g) for some subset S of G, A = ®gGSl (A i ® g) where Si c 5. Since (A\®g) = A\, we find that Cy(A\ g) = H. Thus, it follows that CG(A) = # , because G is an essential extension of Y. This completes the proof. Having solved the (abelian) finite 0-rank case, our next aim is to study the infinite one. The structure of simple modules over abelian groups of infinite rank has been described by D. J. S. Robinson and Z. Zhang in the paper [RZ]. More precisely, in this paper these modules have been considered over the group rings ZG, however this approach with some minor changes is valid for a general

The Structure of Simple Modules over Abelian Groups

23

Dedekind domain. 2.6.Theorem Let D be a Dedekind domain, G an abelian group of infinite 0 rank. (1) If A is a simple DG - module, then t(G/Co(A)) is a locally cyclic pi- group where p = char(D/Anno(A)). (2) Conversely, let I be a prime ideal of D (we allow I to be < 0 >), H a subgroup of G such that r^(GIH) is infinite and t(GIH) is a locally cyclic plgroup where p = char(DIT). If\D\ < ro(G/H), then there exists a simple DG module A with the properties: AnnoiA) = I, CG(A) = H. Proof From Corollary 2.2 it follows that it suffices to show (2). Choose in GIH a maximal Z- independent system of elements {gxH\ X e A}. Put T/H = t(GIH), Y =< T,gx | X e A >. Then YIH = T/Hx(XleA

)

and moreover, G is an essential extension of Y. Choose now a field F in the following way. If / e Spec(D), then put F = DII. \il = < 0 >, then let F be the field effractions of the ring D. In the second case U(F) = £/(£>) X (X^eM < Z„ >)

where < z p > is an infinite cyclic group, \i e M. If D is finite then D = F, so M= 0. If D is infinite, then |D| = \F\ = \F < 0 > | = \U(F)\. It follows that \M\ = \D\. Therefore we can assume that A = M U A i and A/fl Ai = 0. If / * < 0 >, then put R = F[X\ | X e A] be a polynomial ring, and let K be an algebraic closure of the field of fractions of a ring R. If 7 = < 0 >, then put R = F[Xx I X e Ai] and let K be an algebraic closure of the field effractions of the ring R. Let 5 be a complete set of all non-associated prime elements of a ring R. Then \B\ = max{\F\,\A\} = |A| (respectively \B\ = |Ai|). Since R is an unique factorization domain ( see, for example, [KG, Chapter 1 , Theorem 2.20]), in the first case we have U(F(Xx\X s A)) = U(F) x (X t e A < bx >), and in the second case U(F(XX\X e A,))

= U(D) x ( X ^

< z,, >) x (XAEA, < bx >)

(see, for example [KG, Chapter 4, Lemma 1.1]). Since K is algebraic closed then t(U(K)) is a divisible locally cyclic pi- group where p = char(D/T) (see, for example, [KG, Chapter 4, Theorem 6.1]).

24

Simple Modules

Therefore t(U(K)) includes a subgroup which is isomorphic with t{GIH). In other words, there exists a homomorphism 6\ : T —• t(U(K)) with the property KerOi = H. We are extending 9i to a homomorphism 92 : Y —• C/(/Q in the following way. In the first case put 082 = c61 fore e T, gxQi = bx,X € A In the second case put edi = cd\ fore € r, gA02 = 6A, if A e Ai. By construction KerQi = KerOi = H. Since U(K) is divisible, we can extend the homomorphism 82 to the homomorphism 63 :G —• U(K) (see, for example, [FL, Theorem 21.1]). Let u e KerQi. Since G is an essential extension of Tthen there is a number t s N such that uk e Y. Thus 1 = uk93 = uk92, and therefore uk e ATer#2 = H. This means that « e T. Hence « e ATer#i = //, and £er0 3 = H. Put Gi = I1T163. In the first case let A = F\G\~\. In the second case put A = D[G{\. From the definition of 03, it follows that F < D[G{\, and therefore again A = F\G{\. If 0 * a e ^4, then a is algebraic over the field F(Xx\X e A). Therefore a~x e F{XX\X e A)[a]. By the construction of F(XA|A e A) < 7(02), and we have that a -1 e A, what means that ^4 is a subfield of K Thus we can see that A is a simple FG - module. The second case is similar. Obviously, CG(A) = H.

Chapter 3 The Structure of Simple Modules over Some Generalizations of Abelian Groups

Having developed in full the abelian case, we are ready to deduce the structure of simple DG - modules where G is some generalization of abelian group. Nilpotent and hypercentral groups are the first natural extension of abelian groups. Other important generalizations of abelian groups are FC - groups and CC - groups (see definitions below). Thus, the description of simple modules over all these classes of groups will be the aims of this chapter. For the construction of a simple DG - module we will use the following method. In a group G we will choose a "good" abelian subgroup U, then construct a simple DU - module using the results of the previous chapter, and then extend this module to a simple DG - module. We begin with modules over hypercentral groups. Recall that a hypercentral group is a group G having an ascending central series. These well-known groups have arisen as a natural extension of nilpotent (and abelian) groups (see, [RD 10, Chapter 6]. The strategy of the proof is splitting into two cases, depending on the finiteness of 0 - rank. 3.1. Theorem Let G be a hypercentral group of finite 0 - rank, Z = C, (G), T the periodic part ofZ, D a Dedekind domain. (1) If A is a simple DG - module such that CG(A) = < 1 >, then T is a locally cyclic pi- subgroup where p = char(DIAnno(Ay). Moreover, if Spec{D) is infinite, then Anno{A) = P e Spec(D). If DIP is locally finite, then Z = T. (2)IfPe Spec{D), DIP is a locally finite field, p = char{DIP), T = Z is a locally cyclic pi- subgroup, then there exists a simple DG - module A with the following properties: Anno (A) = P,Cc(A) = < 1 >. (3) IfP e Spec{D), DIP is not locally finite, Tis a locally cyclic pi- subgroup where p = char(D/P), then there exists a simple DG - module A with the

26

Simple Modules

following properties: Ann D(A) = P, CG(A) =< 1 >. (4) IfSpec{D) is finite, I is a prime ideal ofD (we allow I to be < 0 >), T is a locally cyclic pi- subgroup where p = char{DII), then there exists a simple DG module A with the following properties: Ann D(A) = I, Cc(A) = < 1 >. Proof Given a simple DG - module A with CG(A) = < 1 >, by Corollary 2.2, T is a locally cyclic/;'- subgroup wherep = char(DII), I = AnnoiA). If Spec(D) is infinite, then by Corollary 1.16, / e Spec(D). Finally, if D/I is locally finite, then T= Zby Corollary 1.21. Let P e Spec{D) and DIP be a locally finite field, Z = T a locally cyclic plsubgroup where p = char(DIP). By Corollary 2.4 there exists a simple DZ module B such that AnnD(B) = P, CZ(S) =< 1 >. Put F = DIP and consider the FG - module B* = B ®FZ FG. In a natural way, we may identity B with an FZ submodule of B*. If Xis a transversal to Z in G, we may write B* = ®x(=xBx. Let A be an FG - composition factor of B*. Then A is a simple FG - module. Since B* is a semisimple FZ - module (i.e. a sum of simple FZ - submodules), we deduce that there is a subset Y c, X such that A is FZ - isomorphic with A o = © x e r Bx. Consider the subgroup CziA) and let x e Y. Then CZ(A) < CziBx) = x~lCz(B)x = CZ(B). This means that CziA) = < 1 >» and, consequently, CaiA) = < 1 > since G is a hypercentral group. The proof of statements (3) and (4) are similar and we omit it Note only, that proving (4) we need to apply Corollary 2.5 instead of Corollary 2.4. 3.2. Theorem Let G be a hypercentral group of infinite 0 - rank, Z = £(G), T the periodic part ofZ, D a Dedekind domain. (1) If A is a simple DG - module such that CciA) = < 1 >, then T is a locally cyclicp!- subgroup where p = chariDIAnnoiA)). (2) Let I be a prime ideal ofD (we allow I to be < 0 >), T a locally cyclic p<subgroup where p = char(DIT). IfG includes an abelian subgroup V such that \D\ < roiV), then there exists a simple DG - module A with the following properties: AnnoiA) = /, CG04) = < 1 >. Proof A straightforward application of Corollary 2.2 gives (1), so it suffices to show (2). Given V as in (2), replacing V by VZ if necessary, we may assume that Z < V. Let Q be a maximal subgroup of Kwith the following properties: Tf]Q = < 1 >, and Q is periodic. Then the periodic part of VIQ is locally cyclic, and UiV/Q) = Tl(T). By Theorem 2.6 there exists a simple DV- module B such that CviB) = Q, and AnnoiB) = I. From the choice of B, it follows that CviB) f\Z = < 1 >.

The Structure of Simple Modules over Some Generalizations ofAbelian Groups

27

Put B* = B ®DV DG, then again B* - (&leSBt where S is a transversal to V in G. Let A be a £)G - composition factor of B*, then ,4 is a simple £>G -module and A ~FZ ®leRBt for some subset R S 5. Moreover, if / * < 0 >, then AnriD(B*) = I, and therefore Anno(A) = /. In this case we consider B*, A as an FG - modules where F = DII is a field. If / = < 0 > we can consider B as a KV module where K is the field of fractions of D. Hence Anno(A) = < 0 >. Also for every / e 7? we have CG(A) n Z < Cz(Bt) = rxCz{B)t

= CZ(B) = < 1 >

This means that CG(A) (~\ Z = < 1 > and, since G is hypercentral, it follows that CG(A) = < 1 >. These two last results are extensions of those of [KSU 3] that deal with the case D = Z. In what concerns Theorem 3.2, we would have to say that the hypotheses appearing there are automatically satisfied by the general case. Indeed, if G is hypercentral and ro(G) is infinite, then, by Maltsev's theorem (see, for example, [RD 10, Theorem 6.36]), G includes an abelian subgroup V of infinite 0 - rank. Since Z is countable, it always satisfies the condition |Z| < n(V). It is time to consider other simple modules, as mentioned. In the next cases generalizing central properties, the groups under consideration will be certain groups with finiteness conditions on their conjugacy classes. Let G be a group, x e G, then put xG = {g'lxg \ g e G}. The subgroup CG{XG)

is normal in G.

Specifically, if X is a class of groups, it is said that the group G have X conjugacy classes or that G is an XC - group, ifGICo{xG) e Xfor all x e G. If X = X is the class of all identity groups, this property describes exactly the abelian groups. Therefore we can consider the class of XC - groups for every class X as an extension of the class of abelian groups. If X = T is the class of all finite groups then we obtain the class of FC groups. Nowadays, the theory of FC - groups form a well developed topic and represents one of the main branches of the Theory of Groups with Finiteness Conditions. If X = C is the class of all Chernikov groups then we come to the class of CC groups, which has been introduced by Ya. D. Polovicky [PY]. Although CC groups are not investigated so far as FC - groups, however their study became very intensive lately (see [AO, FdGT 2, GO 1 - GO 3, GOP, OP 1, OP 2, OPT]). The study of XC - groups for other natural classes X are only at the beginning now (see [FdGK 1, FdGT 1, KL 4, KL 6, KSU 1]).

28

Simple Modules

Let G be a group, X a class of groups. Put XC(G) = {x e G | G/C G (x G ) e

X}.

If /f satisfies some additional conditions, then XC{G) is a (characteristic) subgroup of G. In particular: A class of groups X is called a formation if it satisfies the following conditions: (Fl) IfG e X,Hisa normal subgroup ofG, then GIH e X; (F2) If Hi, H2 are normal subgroups of a group G such that GIH\, G/H2 e X, then G/(Hi n H2) e * . It is very common to express the above properties by saying that X is Q-closed (closed under taking images) or Ro-closed (closed under taking finite subdirect products), respectively. In short, X= QR 0 .Y. Note that A" is a formation of groups, then XC{G) is a characteristic subgroup ofG. The subgroup XC(G) is called the XC - center of the group G. A group G is an XC - group if and only if G = XC{G). Starting from XC - center we can construct the upper XC - central series of a group G: < 1 > = Co < Ci < ... Ca < Ca+l < ... Cy where C\ = XC(G), Ca+i/Ca = XC{GICa\ a. The last term Cr of this series is called the upper XC - hypercenter and denote XCX{G); the terms Ca are called the XC - hypercenters. If XC'00(G) = G, then the group G is called XC - hypercentral, if y is finite then G is called XC - nilpotent. If X = X is the class of all identity groups then we obtain the hypercentral groups. If X = T is the class of all finite groups, then we obtain the FC hypercentral groups. In this case we can think about the upper FC - central series and use the designation FCa(G) for the terms of this series, so FC{G) = FC\{G) is the FC - center of G, FCX(G) is the upper FC - hypercenter of the group G. 3.3. Lemma Let Xbe aformation of groups, G a group,P = XC{G), Q = XCoo(G), H a G - invariant subgroup of Q. If H * < 1 >, then Hf\P < 1 >. Proof

Let us denote by

*

The Structure of Simple Modules over Some Generalizations ofAbelian Groups

29

< 1 > = Co < Ci < ... C« < Ca+l < ... Cr = Q the upper XC - central series of G. There is an ordinal a such that Hf]Ca * < 1 >. Let B be the least ordinal with this property. By definition B is not a limit ordinal, so that HC\ Cp-\ = < 1 >. Let 1 * x e H(~) Cp,X = < x >G . Since x Cp-i £ CplCp-u GICG{XCp.yICp-X) e X. L e t g e C G ( X Q _ i / Q - . , ) . Then [g,X] < Cp-i. On the other hand, H is a normal subgroup, so that \g,X] < //, and [g,X] . This means that CG{XCp-\ICp-\) < CG{X), in particular, since X is a formation, GICG(X) e A". .Hence X < XC(G). • 3.4. Corollary Let X be a formation of groups, G an XC - hypercentral group. If A is a non-identity normal subgroup ofG then A f] XC(G) # < 1 >. To deal with the most common cases, the following results are very useful. 3.5. Lemma Let R be a ring, G a group, H a normal subgroup ofG, A an RG module, B an RH - submodule of A. (1) If g £ G, then Bg is an RH - submodule. Moreover, if B is an noetherian {respectively artinian) RH - submodule, then Bg is a noetherian(respectively artiniari) RH - submodule. IfB is a simple RH - module, then Bg is also a simple RH- module. (2) If A is a simple RG - module, B a simple RH - submodule then A = © „ e S Bg for some subset S c G. In particular, A is a semisimple RH - module. Proof

(1) Since H is normal in G, gRH = RHg for each g e G. Therefore {Bg)RH = B{gRH) = B{RHg) = (BRH)g = Bg.

Let {C„ | n e N} be an ascending (respectively descending) chain of RH submodules of Bg. Then {C„g~x | n e N} is an ascending (respectively descending) chain of RH - submodules of B. If B is a noetherian (respectively artinian) RH - submodule, then there is a number m such that Cmg~x = Cm+ng~x for all n e N. It follows that Cm = (Cmg~l)g = {Cm+ng~x)g = Cm+„ for all n e N. Let B be a simple RH - submodule, C a non-zero RH - submodule of Bg. If we assume that Cg~l = < 0 >, then we will get that C = < 0 >. This means that Cg~l = B, thus C = (Cg _1 )g = Bg. In other words, Bg is a simple RH submodule. (2) Put C = XI GB8- T h e n c i s a n RG~ submodule of A. Since C * < 0 >, C = A. Every summand Bg is a simple RH - submodule, therefore A = © ^ ^ f i g for some subset S^G. 3.6. Proposition [HB 3, ZD 1] Let R be a ring, G a group, H a normal subgroup of G, A a simple RG - module. If\G:H\ is finite, then A includes a

30

Simple Modules

simple RH - submodule B and A = © „ 5 Bgfor some finite subset S such that \S\<\G: H\. Proof By Lemma 3.5, it suffices to show that A includes a simple RH - module. Let S be a transversal to H in G. If T c S and B is an RH - submodule of A; we will write T ~ B to indicate that BT = ^2leTBt = A, but BT\ *• A for every proper subset T\ of T. Since A is a simple RG - module, we note that A = BG = BS. Thus, for an RH - submodule B, there exists T c S such that T ~ B. The plan of the proof is to show the existence of an RH - submodule U of A and a subset So £ S, such that So ~ V for any RH - submodule V of U. It follows that U is simple. The construction of this pair (U,So) will be carried out inductively as follows. Put B\ = A and choose S\ S S such that S\ ~ B\. Suppose there exists a proper RH - submodule B2 < B\ such that S\ * B%. For this B2 we may choose a subset 52 £ S such that £2 ~ B2. Proceeding in this way, we can construct a descending chain of non-zero RH - submodules B\ > B2 > ... > Bt and a collection S\,S2,— ,Sj of subsets of S, which satisfy Bj * BJ+\, Sj ~ BJy Sj * BJ+\,j < i. We claim that the subsets Sj are all distinct. Suppose that there are indexes k and m, k < m, such that Sk = Sm. Then Bm is a proper submodule of Bk,Sk ~ Bn and Sk = Sm ~ Bm. Then Bk+iSk > BmSk = BmSm = A. However, if 7Ms a proper subset of Sk, then Bk+t T < BkT * A. Summing up, Bk+iSk =A but Bk+\T±A, which means that Sk ~ Bk+\; a contradiction, which proves our claim. Hence, the subsets Sj are all distinct. Since S is finite, we cannot produce infinitely many of them, so there exists a number n e N in which the process finishes. This means that the RH - submodule B„ and the subset S„ satisfy the following condition: ifC is a proper RH - submodule ofB„, then S„ ~ C. Evidently, the above condition express that CS„ = A but CT * A for every proper subset T of S„, so this pair (B„,S„) satisfies the required conditions mentioned above and our next goal is to finish the proof of the simplicity of B„. Suppose that S„ = {x\,...,xt} and define Kj = B„Xj P\ QZ ( i B„xi), 1 <j < t. By Lemma 3.5 each Kj is an RH - submodule of A, so every Lj = KjXj1 is an RH - submodule too. Since Lj < B„, LjX> < B„Xj, whenever / * / Thus LjXj = Kj < ^.^.BnXj, and it follows that LjS„ = ^l and so K}•• = < 0 >. This implies that ,4 = B„S„ = ® 1 J i 9 B ^ , . Under these conditions, assume that B„ includes a proper non-zero RH - submodule C. Then CS„ = X K , < , £ * ' * S K K , ^ " * ' = ^ ' which contradicts with the choice of B„. This contradiction shows that B„ is a simple RH - submodule, and all is proved.

The Structure of Simple Modules over Some Generalizations ofAbelian Groups

31

3.7. Corollary Let Rbe a ring, G a group, H a normal subgroup of finite index. If A is a semisimple RG - module, then A is a semisimple RH - module. Note that this corollary extends on the infinite case the famous Clifford's theorem [CA]. 3.8. Lemma [FdeGK 3] Let F be afield, G a group, A an infinite simple FG module such that CG(A) = < 1 >. Let P be aG - invariant elementary abelian p subgroup of FC{G), p is prime. Then char F * p and P includes a subgroup J such that \PU\ = p and Corea{J) = < 1 >. Proof Let M b e a minimal G - invariant subgroup of P. Since P < FC(G), M is finite, and therefore, H = CQ{M) has finite index in G. By Proposition 3.6 there exists a finite subset X £ G and a simple FH - submodule B such that A = @xeXBx. Since M i s normal in G and CQ{A) = < 1 >, CA(M) is a proper FG - submodule of A, so that CA(M) = < 0 >. This means that M $ CH(B) and Corollary 2.2 implies that charF =£ p. Let K be a finite G - invariant subgroup of P,0 * a e A,A i = aFK. Since K is finite, dimpA \ is also finite. Therefore A \ includes the simple FK - submodule B. FutC = BFP = '£xepBx. Let £ be a local system of P consisting of finite non - identity G - invariant subgroups of P, including K,L e C. Since P is abelian, the equation C = ^xePBx shows that CL{B) = CL(Q. Put E = BFL. By Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]), E = ® 1 < J < r £ ; where Et is a simple FL submodule. By Corollary 2.4 LICL{E\) is cyclic, and therefore \LICL(E\)\ = p. The equation Q ( # ) = CL(C) implies that CL(C) = CL{E\). This means that \LICL(C)\ = p. Since it is true for each L e C, \P/CP(C)\ = p. Put J = Cp(C), U = Corea(J). Let u e U. Since A is a simple FG - module, A = BFG. Therefore for every element d e A we have d = 2j 1<1
Zi<,, = Zi<,<>*< = dIt follows that U < CG(A) = < 1 >. 3.9. Lemma [KO 3] Let G be a CC - group. Then G includes a normal subgroup U satisfying the following conditions. (1) U = XA.EA'S'A where S^ is either a finite minimal normal subgroup ofG, or Sx is an infinite cyclic normal subgroup ofG, X e A;

Simple Modules

32

(2) IfH is a non-identity normal subgroup ofG, then HD U * < 1 >. Proof Let S be a subgroup generated by all minimal G - invariant subgroups of FC{G). Since every minimal normal subgroup of a CC - group is finite [PY], S is generated by all finite minimal normal subgroups of G. There exists a set M such that S = X^SMS^ where S^ is a finite minimal normal subgroup of G, /J. e M (see, for example, [RD 9, Lemma 5.23]). If this S satisfies the condition (2), then it suffices to define U = S. Otherwise, there exists a non - identity normal subgroup H such that Hf)S = < 1 >. Let 1 * h e H,Q = < h >G . Since G is a CC group, we can conclude that either Q is a Chernikov subgroup, or Q includes a G invariant Chernikov subgroup R such that QIR is infinite cyclic [PY]. If we assume that R * < 1 >, then R includes a finite G - invariant minimal subgroup, that is R n S * < 1 >. This means that R = < 1 >, i.e. Q is an infinite cyclic subgroup. Put Si = Sx Q.lf Si satisfies (2), then define U = Si. If Si does not satisfy (2), then in order to finish the proof we just need to proceed in the same way using transfinite induction. The subgroup U of the CC - group G occurring in the above result is said to be a quasi-socle ofG and denoted by Qsoc{G), if it does not lead to any mistake. In contrast with the usual socle, the existence of a quasi-socle in a given group is doubtful, although every CC-group has at least one. In this case, if G is a CC-group, then Soc(G) < Qsoc(G), where Soc{G) is the ordinary socle ofG. Let Soc(G) = XASASA where Si is a minimal normal subgroup of G, A e A. Put A„A = {A e A | Si is abelian }, and SoCabiG) — XlcA^Sl 3.10.Theorem Let G be an infinite CC - group, D a Dedekind domain, A a simple DG - module such that CG(A) = < 1 >. Then (1) Socab{G) is apt- subgroup wherep = char{DIAnno{A)); (2) Socab(G) includes a subgroup Q such that Socab{G)/Q is a locally cyclic group and Core G(Q) =< 1 >; (3)IfSpec(D) is infinite andro(G) is finite then Anno{A) = P e Spec(D); (4) IfAnnD(A) = P e Spec{D) and DIP is a locally finite field and r0(G) is finite, then every quasi - socle ofG coincides with the socle ofG. Proof Ifp = char{DIAnno(A)), then by Lemma 3.8 Socab{G) is ap>- subgroup. On the other hand, let Srbe a Sylow r - subgroup of Socab{G), r e TI(Socai,(G)).

The Structure of Simple Modules over Some Generalizations ofAbelian Groups

33

Then Sr includes a subgroup Qr such that \Sr/Qr\ = r and CoredQr) = < 1 >, by Lemma 3.8. Put Q = Xren(5ocat(G)) Qr -Then Socab{G)IQ is locally cyclic and CoreaiQ) = < 1 > . Since G is a CC-group, the normal closure of every finite subset of G is Chernikov-by-(finitely generated abelian) [PY]. It follows that every finitely generated subgroup of G is central-by-finite. In particular, a CC-group G is locally (polycyclic-by-finite). Then (3) follows from Corollary 1.16. Finally we will prove (4). Assume that some quasi - socle of G does not coincide with the socle of G. This means that G includes a normal infinite cyclic subgroup C. Put H = CG(C); so that \G : H\ < 2. Proposition 3.6 yields that A includes a simple DH - submodule B such that A = B © Bx for some x e G\H. If we assume that C n C//CB) * < 1 >,

c n cH(Bx) = cn (x~l)cH(B)x *< I > . It follows that C n CH{B) = < 1 >. In this case C s C(C„(5))/C„(fl) <

RHICiffl),

being the latter a periodic group, by Corollary 1.21. This final contradiction shows that our assumption does not occur and proves (4). A group G is called hyperfinite, if it has an ascending series of normal subgroups with all factors finite. Obviously, the hyperfinite groups are exactly the periodic FC - hypercentral groups. In particular, the socle of a hyperfinite group is non-identity. 3.11. Theorem Let G be a hyperfinite group, D a Dedekind domain, A a simple DG - module such that Ca(A) = < 1 >. Then (1) Socab{G) is a pi- subgroup where p = char(D/AnnD(A)); (2) Socab(G) includes a subgroup Q such that Socab{G)IQ is a locally cyclic group and CoreG{Q) = < 1 >; (3)IfSpec(D) is infinite, thenAnnD{A) = P e Spec(D). The proof repeats on the whole the proof of the previous theorem. The following question naturally arises: for what CC - groups G there exists a simple module A such that CG{A) = < 1 >? 3.12. Lemma [KO 3] Let F be a field, S = XX^ASX where Sx is a finite non-abelian simple group for any X e A. Then there exists a simple FS - module

Simple Modules

34

AsuchthatCs(A)

= < 1 >.

Proof Let y be the type of A. Proceeding by transfinite induction, we are going to show that S has an ascending series of normal subgroups < 1 > = Co < Ci < ... Ca < Ca+i < ... Cy = S such that Ca+i = Ca x Sa+i where Sa+i = Sx„, for some Aa+i e A, a < y. We will apply induction on a. Let a = 1. Since C\ is a simple finite group, the Lemma follows from Lemma B.10.2ofthebook[DH]. Let a > 1. Suppose that the Lemma is proved for all ordinal j3 < a. Let a = P + 1 for some p. Then there is a simple FCp- module Vsuch that CcpiV) = < 1 >. Also there is a simple FSp+i - module B such that Cs^ (B) = < 1 >. Let U = V(BF B. Then U = ®xeX Vx for some subset X c Sp+\. We can consider U as an F(Cp xSp+\) - module by Corollary B. 1.12 of [DH]. Let A be a composition F(Cp+\) - factor of U. Then ,4 is a simple F(C/j+i) - module. Let L = Cc^(A). Since U is a semisimple F(Cp) - module, A = (&xeX Vx for some I i E l and CcpiV) = < 1 >, we can conclude that L f] Cp = < 1 >. Similarly, U = ®ysYBy for some subset Y ^ C^+i, so that U is a semisimple FSp+i - module. It follows that A = ® > , e7i By for some subset Y\ c y. Since Cs^, (B) = < 1 >, L fl S/j+i = < 1 >. In particular, [L,Sp+\] = [L,Cp] = < 1 >. This means that L < C{Cp+\). However Cp+\ is a direct product of finite non - abelian simple groups, so that its center is trivial. Hence L = < 1 >, that is CQM (A) = < 1 >. Let a be a limit ordinal. For every p < a there is a simple F(Cp) - module Ap such that Ccp{Ap) = < 1 >. It follows from the previous proof that we can choose the module Ap such that the inequality Ap <Ap+\ holds for any p < a. Put Aa = lim{Ap \p < a}. Then Aa = [Jp. Then there is the least ordinal 8 such that E fl As * < 0 >. It is clear, that 8 is not limit. Then E f] As is a non - zero FCs- submodule, that is As = E f] As and As < E. By Lemma 3.5 As+l = © z e z ^ z for some (finite) subset Z £ Cs+i • Since £ is a FCs+i - submodule, the inclusion As < E implies Asz < E for each z e Z. It follows that As+i < E. Ordinary induction proves that Ap < E for any p
The Structure of Simple Modules over Some Generalizations ofAbelian Groups

35

Then

Denote L = CcMY Lf)Cp < CCMP) = < 1 >• This means that L = < 1 > because Ca = \Jp. 3.13. Theorem Let D be a Dedekind domain, G a CC - group of finite 0 - rank, Q a subgroup ofSocab(G) such that Socab{G)IQ is locally cyclic and CoreG{Q) = < 1 >. (1) Assume that Spec(D) is infinite, P e Spec(D), DIP is a locally finite field, and Socab(G) is a pi- group where p = char{DIP). If every quasi - socle of G coincides with the socle ofG, then there exists a simple DG - module A satisfying the following conditions: Ann D{A) = P,Ca(A) = < 1 >. (2) Assume that Spec{D) is infinite, P e Spec(D), DIP is not locallyfinite.If Socab{G) is a p<- group where p = char(DIP), then there exists a simple DG module with the following conditions: Anno(A) = P, CQ(A) = < 1 >. (3) Assume that Spec(D) isfinite,I is a prime ideal ofD (we allow I to be < 0 >). IfSocab(G) is a pi- group where p = char(D/I), then there exists a simple DG- module A with the following properties: Anno(A) = I andCaiA) = < 1 >• Proof Let S = Soc(G), R = Socab(G). Then S= RxT where T is a direct product of finite simple non-abelian groups. Assume that Qsoc(G) = Soc(G), P e Spec(D), F = DIP is a locally finite field and Socab(G) is a p<- subgroup where p = charF. By Corollary 2.4 there exists a simple FR - module B such that CR(B) = Q. By Lemma 3.12 there exists a simple FT- module C such that CT(C) = < 1 >. Put U = B ®F C and consider U as F(R x T) - module ( [DH, Corollary B.1.12] could be useful at this point). Let E be an FS - composition factor of U, then £ is a simple FS - module. Let L = Cs(E). By Lemma 3.5 U = ®xeJ( Cx for some subset X c S. It follows that E = ® x e X Cx for some subset Xi c X. In turn, it follows that L D T = CT(C) = < 1 >, i.e. L < CS(T). Since T is a direct product of non - abelian finite simple groups, Cs(T) = R, and L < R. Similarly, by Lemma 3.12 there is a subset F c J such that U = ® 6yj8y, and E = ® eY By for some subset Y\ c Y. This proves the equality L = CR(B) = Q. Form V = E ®FS FG and take A to be an FG - composition factor of V. One more time, it is possible to write V'= © z e Z £ z for some subset ZcG and A s ® z s Z Ez where Z\ £ Z Then A is a simple FG - module. If H is a non identity normal subgroup of G then Hf\ Qsoc(G) = HC\Soc(G) * < 1 >. If we assume that CG(A) * < 1 >, then Cs(A) * < 1 >. If g e G we have that CS(A) < Cs(Eg) = g-lCs(E)g = g-lQg. Since it is true for every g e G,Cs(A) < f] CG(A) = < 1 >.

8

GGQ

= < 1 >. Consequently,

36

Simple Modules

Let F = DIP be not locally finite, S\ = Qsoc(G). Then S\ = S x S2 where S2 is a direct product of G - invariant infinite cyclic subgroups. By Corollary 2.4 there exists a simple F(R x S2)- module B such that CRXS2(B) = Q. The rest of the proof is similar. lfSpec(D) is finite, then instead of Corollary 2.4 we must use Corollary 2.5. The case in which the 0-rank of the group is infinite runs in a similar way, although we need to impose certain additional hypotheses and proceed very carefully. 3.14. Theorem Let D be a Dedekind domain, G a CC - group of infinite 0 rank, Q a subgroup ofSocab{G) such that Socab(G)IQ is locally cyclic and CoredQ) = < 1 >. Let I be a prime ideal ofD (we allow I to be < 0 >). If Socab(G) is a pi- group where p = char(DII) and G includes an abelian subgroup Vsuch that \D\ < ro(V), then there exists a simple DG - module A with the following properties: Ann D(A) = landCdA) = < 1 >. Proof Let Z be an upper hypercenter of G, S = Qsoc(G), R = Socab(G). In this case we may write S = RxTxU where T is a G - invariant subgroup of G such that T is a direct product of non - abelian finite simple groups, U is a direct product of G - invariant infinite cyclic subgroups. It is well-known that the commutator subgroup [G,G] of a CC - group G is a periodic group (see [PY], for example), therefore U < C(G). If M is a minimal normal subgroup of G, then either M< Z or M f l Z = < 1 >. This simple remark allows us to decompose R = Ri x R2, where R\ = Rf]Zand R2 is a G - invariant subgroup with property R2 n Z = < 1 >. On the other hand, by [FdGT 2, Theorem 3.2], G/Z is a periodic group. It follows that ro(K) = ro(Vf)Z) and so we may assume that V < Z. Also Ri U is central, so we may suppose that R\U < V. Since Q is core-free, then QV\Z = < 1 >. Let Q\ be a maximal periodic subgroup of Fwith the property R\ ft Q\ = < 1 >. Then the periodic part of VIQ\ is locally cyclic and Yl(VIQ\) = Tl(Ri). Moreover, [Z,TR2] = < 1 >, so that V\ = VxR2 is abelian. By Theorem 2.6 there exists a simple DV\ - module B with the properties Anno(A) = I and CVl(B) = QxQi. Now, we define the field F as follows. If / * < 0 >, we simply put F = DII. Otherwise F will denote the field of fractions of D. By Lemma 3.12 there exists a simple FT - module C such that CT(Q = < 1 >. Put Si = V\ x T and B\ = B®FC. Then S <S\ and we may think of Bi as a FS\-module (see [DH, Corollary B.1.12]). By Lemma 3.5 there is a subset Y ^ Si such that Bi = ® r 5 y . Let £ be a FSi - composition factor of B\, and we have that E = © SKi By for some Yi c Y. It follows that E is simple DSi - module. As in the proof of Theorem 3.13, one easily deduces that Cs,(E) = Q x Q\. Put E* = E ®£>s, DG. Then E* = @xsXEx

where X is a transversal to Si in

The Structure of Simple Modules oxer Some Generalizations ofAbelian Groups

37

G. Let A be a DG- composition factor of E*. As usual, A becomes a simple DG module which decomposes A = Q)xeJ( Ex where X\ c X. Let L = CG(A). Then L is a normal subgroup of G. Suppose that L C\ S = Cs(A) * < 1 >. By A = ®xeJ(Ex we obtain that LnS
= CS(E)

in particular, L 0 S < Q x Q\ For each g e G w e have

ms = g -'(in% < g-'(e>< 2i)g, so that L n 5 < CoreG(Q *• Q\) = W. Then

» , nz<(6'
wnz = g-\wr\Z)g, then CoreG(Q\) (1 f(G) * < 1 > by Lemma 3.3. In particular, Q{ n C(G) * < 1 > . However, this contradicts the election of Q\. Consequently, CoreaiQi) = < 1 >, so that and WD Z = < 1 >. Since W < Socab(G), W includes a finite minimal G invariant subgroup M t . Since WnZ = < 1 >, Co (Mi) * G. On the other hand, Wr)R2<(QxQi)nR2

= Q(Qif)R2) = Q,

and

wnRi =g~l(wnR2)g. Since Mi < R, M1R2/R2 < R/R2 = R1R2/R2 < ZR2/R2. Hence for every g e G,x e Mi, we have [g,x] e R2. But Mi is normal in G, so that \g,x\ e A/i n « 2 = < 1 >• This means that Mi < £(G), i.e. C G (Mi) = G. This contradiction implies that W = < 1 >, so that Z, n S = < 1 >, and CG(A) =

38

Simple Modules

< 1 >.

3.15. Theorem Let D be a Dedekind domain, G a hyperfmite group, Q a subgroup of Socab(G) such that Socab{G)IQ is locally cyclic and CoredQ) = < 1 >. (1) Suppose that Spec{D) is infinite, P e Spec(D) andSocab{G) is ap>- group where p = char{DIP~). Then there exists a simple DG - module A satisfying the following conditions. Ann D{A) = P,CG(4) =< 1 >. (2) Suppose that Spec{D~) isfinite,I is a prime ideal ofD {we allow I to be < 0 >). If Socab(G) is a pi- group, where p = char{DIT), then there exists a simple DG - module A with the following properties: Anno(A) = I and CG(A) = < 1 >. The proof of this theorem just repeats in its main details the proof of Theorem 3.13. We conclude with some remarks. Firstly, in the case D = Z, the Theorems 3.10, 3.13, 3.14 come from [KO 3]. We should recall that the addition assumptions are automatically satisfied in this case. As we alredy mentioned, the factor-group G/£X(G) is periodic. Let G be a CC - group of infinite 0 - rank. Using Mal'tsev's theorem (see, for example, [RD 10, Theorem 6.36]) we obtain that fa>(G) includes an abelian subgroup V of infinite 0 - rank. Since Z is countable the condition |Z| < ro{V) is always realized . Finally, we would like to mention that for locally soluble FC - groups G the structure of simple FG - modules have been studied in [FdeGK 3]. The following question naturally arises in connection with the main results of this chapter. Question 3 Let F be afield. What is a soluble {metabelian) group G such that there exists a simple FG - module and Ca{A) = < 1 > ?

Chapter 4 Complements of Simple Submodules

Let A be a module over a ring R, B a submodule of A. We say that B has a complement in A {or B is a complemented submodule) if there is an R submodule C such that A = B © C. In the theory of modules over the group ring RG the following problem arises very often. For what submodule B there exists a complement! In particular, When a simple submodule B has a complement ? As we already mentioned many times, the case of modules over group rings is especially important for group-theoretical applications. The famous Maschke's Theorem is one of the first results about existence of complements of submodules over finite groups. We are not going to give a survey of main results in this area. We can just mention some references related to them ( see, for example, the survey [KZK] and the papers [DZ 1 - DZ 6], [DT], [HB 2 - HB 5], [HBT], [KN], [KPS 1, KPS 2], [ZD 1], [ZD 2], [ZD 5], [ZM]). In this chapter we are going to obtain some very specific partial results useful for the study of just non-Af-groups. Fortunately, we could collect them together in a general way which was satisfactory for our purposes. Consequently, the results will be stated in such a way that we were able to suppose that the underlying groups belong to some given class of groups and the proofs will make use of the general language of classes of groups and the relations among them. 4.1. Theorem

Let X- be a formation of groups, such that X(X fl T) = X.

39

40

Simple Modules

Suppose that G is both an FC - hypercentral and an XC-hypercentral group. Let K be afield, A a KG - module, B a KG - submodule of A satisfying the following conditions: (/) B andAlB are simple KG - modules; (ii) G/Ca(A/B) e XbutGICG{B) £ X. Then there exists a KG - submodule C of A such that A = B®C. Proof Obviously, we can assume that CG{A) = < 1 >. Since X is a formation and GICG{B) £ X, we have G £ X. This and GICG{AIB) e X give CG(A/B) *< 1 >. Since G is both FC - hypercentral and A^C-hypercentral, G is (^n-TDC-hypercentral. Hence if Q is the (Xn?)C - center of G, then Q n CG(A/B) * < 1 > by Lemma 3.3. Consider a non-identity element > of this intersection. Thus < y >G is central-by-finite and GICG{< y >G ) e X f\ F. Put Y = < y >G and H = CG{Y). We consider the family S = {E | E is a KH - submodule of A such that E $ B}. Clearly A £ S, so S * 0. By [WJ 1, Theorem A], A is an artinian KH-modu\e and hence S has a minimal element, say M. Let u s Y. Note that M(u- 1) =KH MICM(U). Since [H,K] = < 1 >, we can conclude that M{u- 1) and CM(U) are KH - submodules. Since u e CG{AIB), we also have

M(u-\)


Lemma 3.5 implies the decomposition B = ®, e 7 - Vt, where V is a simple KH submodule and T is a finite subset of G We claim that HICH(V) £ X; for, e otherwise every HICHW) X and, since ^ is a formation, we would have that HICH{B) e X Since G / # e # n .F and # = X(Xn P), we would be able to deduce that G/CG(B) e X(Xn T) = X, contradicting (ii). Thus H/CH(V) £ X, as claimed. Given u e Y, suppose that CM(U) < B. Then MICM{U) has a finite KH composition series in which only one factor is ^-central. On the other hand, since M/CM(U)

=KH M(U-

1) < B, M{u-Y)

can be written as M ( « - 1) = 0 / e 7 - Vt

where T\ c r . This indicates that M(u- 1) has no-central X -factors in any AW-composition series, because HICniY) £ X . This contradiction shows that CM(U) G <S, and, by the choice of M, CM(U) = M for every u e Y. It follows that M < CA(Y). In particular, CA(Y) $ B. Since Yis a normal subgroup of G, CA{Y) is a AG-submodule of ^ , so that, by simplicity of B, we have that either B < CA{Y) or B n CA(Y) =< 0 >. In the first case, by simplicity of AIB, A = CA(Y), C = CA{Y),

and so Y < CG(A) = < 1 >, a contradiction. Therefore, if we put then we get A = B © C, as required.

We need the following result about Dedekind domains.

Complements of Simple Submodules

41

4.2. Lemma Let Dbe a Dedekind domain, P e Spec(D), n e N. (1) All non-zero ideals of DIP" are DIP",PIP",... ,P"-l/P". (2) The D - modules DIP andP"~xIP" are isomorphic. (3) Ify e P\P2 then P =yD + P". Proof (1) follows from [KG, Lemma 3.3.4], (2) follows from the proof of Corollary 3.3.15 of the same book [KG]. Finally.consider the ideal yD + P". Lemma 3.3.4 from [KG] implies that yD + P" = Pk for some 1 < k < n. Since y £ P2,k = 1 In this way (3) is proved. 4.3. Corollary Let D be a Dedekind domain, X a formation of finite groups, G an XC-hypercentral group, A a DG- module and B a DG - submodule satisfying the following conditions: (i) B andAlB are simple DG-modules. (ii) GICG{AIB) e XbutGICG{B) <£ X. Then there exists an DG-submodule C ofA such that A = B © C. Proof Suppose first that AnnD(B) = P * < 0 >. Then P e Spec(D). We compare P with Q = Anno(A/B). Consider first the case when Q * < 0 >. If Q * P, then it suffices to consider the g-component C of A to obtain the direct decomposition A = B © C, as required. lfQ = P, we have that AP2 = < 0 > and by Lemma 4.2 there exists an element^ e P such that P = yD + P2. In this case, the mapping

B. If Im

, we have that Im

, that is, AP =< 0 >. This allows to us to think of A as a £G-module, where F = AIP, and then to be able to apply Theorem 4.1. Thus, it only remains to consider the case in which Q = < 0 >. In this case AIB is a £>-torsion-free and divisible module, so that B = to(A). Since Anno(B) * < 0 >, there exists a D-submodule E such that we may decompose A = B@ E (see, for example, [KI]). But E, being isomorphic with AIB, is also divisible, so that AP = EP = E. This implies that £ is a DG-submodule, because AP has this structure. Furthermore, if Anno{B) = < 0 >, we find out that B is D-divisible. As above, we have a D-lineal decomposition A = B © E\, for some Z)-submodule E\ (see, for example, [SV, Proposition 2.10 and Theorem 2.15]). If AnnD{AIB) *< 0 >, it follows that E\ is the D-periodic part of A, and so E\ itself is a DG-module. If AnnD{AIB) =< 0 >, AIB becomes D-divisible too and we may consider A as a A^G-module where K is the field of fractions of D, which allows to apply Theorem 4.1 again. Our next consequence will need a notion which fairly extends that of the simple module. As usual, in the conception of these ideas, it comes from the ring

42

Simple Modules

of integers Z. Let R be an integral domain and take G to be an arbitrary group. We say that the RG-module A is R-irreducible if A satisfies the two following conditions: (i) A is R-torsion-free. (ii) For every non-zero RG-submodule B of A, the factor-module AIB is R periodic. Thus, for R = Z, Z-irreducible = torsion-free rationally irreducible, the mentioned idea from which this concept comes. Obviously a simple #G-module satisfies the above condition (i), (ii). As usual, the concept can be characterized through the extension ofscalars. That is, if K is the field effractions of J?, an R-torsion-free RG-module is R-irreducible if and only if the KG-module A ®RK is simple. 4.4. Corollary Let D be a Dedekind domain, X a formation of finite groups, G an XC-hypercentral group, A be a DG - module and B a DG - submodule of A satisfying the following conditions: (i) A is a D-torsion-free module; (ii) B is D-irreducible; (Hi) G/CG(A/B) e XbutGICG(B) £ X. Then there exists a non-zero DG-submodule C ofA such that B C\C = 0. Proof Put £ - A <8>D K, where K is the field of fractions of D and think of £ as a ATG-module. Define B\ = B <£>£> K, so that B\ is a simple .KG-module and CGOBI) = CG(B).

Given a e A\B, put E\= aKG + B\. Since CG(EIBi) = CG(AIB), G/Ca(Ei/Bi) eX From X c T it follows that dimK(E\IB{) is finite. In particular E\IB\ includes a simple /^G-submodule, say EilB\. It readily follows that the modules Ei,B\ and EilB\ satisfy all the conditions of Theorem 4.1. Hence there exists a /TG-submodule C\ < £2 such that £2 = B\ © C\. Since E is a D-essential extension of A, C = C\ f\A * < 0 >. But B D C = < 0 >, so C is the required module. In the study of just non-X-groups it is very important to know when a group G splits over its Af-residual. Note that theX-residual of a group G is the intersection of all normal subgroups of G giving a quotient an A'-group. We recall that a group G is splittable over its normal subgroup H (or that G splits over H) if H admits a complement (as a subgroup) in G, that is, if there exists a subgroup Q such that G = HQ and H(~)Q = < 1 >. We indicate this as G = H X Q. If all the complements to H in G are conjugate (in G), we say that G conjugately splits over H. Often a monolithic just non-A'-group conjugately splits over its monolith. It could be proved for each particular class X. However, there exists a very important

Complements of Simple Submodules

43

result, which is happen to be univesal for all nedeed cases. It is the following Robinson's theorem. 4.5. Theorem [RD 22] Let G be a group with an abelian normal subgroup A satisfying the minimal condition on its G-invariant subgroups, H a normal subgroup ofG satisfying the following conditions: (/') H > A, HI A is locally nilpotent; (ii) the upper FC - hypercenter ofGICtiiA) includes HICH(A)\ (at) AH an) =< i >• Then every over group L > A ofA conjugately splits over A. We omit the proof of this theorem, since it requires a very specific technique, straying far from our goals. One can find this proof in the paper [RD 22] and also inthebookfAFdeG].

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Part II Just Infinite Modules

As we already noted, another type of modules which play a relevant role in the study of just non-^-groups are the just infinite modules. Because for some noetherian modules it is possible to make a reduction to just infinite modules, the investigation of just infinite modules is important by itself. These modules arise by the following way. Let A be an infinite noetherian module over a ring R, M. = {C\C is an R- submodule of A such that AIC is infinite} Then M has a maximal element M. Put V = AIM. If U is a non-zero submodule of V, then VIU is finite. The following two cases are possible here: (1) V includes a non-zero simple submodule S; (2) the intersection of the family of all non-zero submodules of Vis zero. In the first case S is infinite and VIS is finite; therefore, it could be reduced to the cases of infinite simple modules and finite modules. Moreover, for some types of rings V is exactly a simple module; for example, if R = ZG, where G is a hypercentral, (even FC - hypercentral, group). Hence the second case here is the main. Let R be a ring. An R - module A is said to be a just infinite if it satisfies the following conditions: (JI I) for every submodule B of A a factor - module A/B is finite; (JI2) the intersection of the family of all non-zero submodules of A is zero. These modules have been introduced by D. J. S.Robinson and J. S. Wilson in their fundamental paper [RW], where the groups with polycyclic proper factor-groups have been studied . Since a simple ZG - module over a polycyclic group G is finite ( see, for example, [PD 1, Theorem 12.3.7]), we recall that the

45

46

Just Infinite Modules

condition (JI 2) is optional for a just infinite ZG - module. The further researches have shown that these modules play an important role in the study of just non-^-groups and for many other important classes X of groups, in particular, for all classes of groups X, which will be considered in this book. Note that just infinite modules play a crucial role also in the study of the generalized soluble groups with the weak maximal and minimal conditions for normal subgroups and in some other important investigations.

Chapter 5 Some Results on Modules over Dedekind Domains

For the study of a module A over a group ring DG where D is a Dedekind domain the structure of A as a Z)-module plays a very important role. In this chapter we have collected some results about modules over Dedekind domains, which are necessary for the study of just infinite modules over DG. In particular, we consider an analogy of the important concept of a p - basic subgroup in Abelian Group Theory. We have no found analogies of this concept for modules over Dedekind domains in the journal literature. In this case, as well as in all remaining cases, we followed the analogy from Z to D. Our purpose is to prove the existence of P- basic submodules in torsion-free modules and, using this, to prove some results about torsion-free D-modules, which are necessary for us. Thus, this chapter hass a technical character. Let D be a Dedekind domain, P a maximal ideal of D. Then the D - modules DIP" and P/P"+l are D - isomorphic (see, for example the proof of Corollary 3.3.15 from [KG]). In particular, DIP" is embedded in D/P"+l, n e N. Therefore we can consider the injective limit of the family {DIP"\n e N}. Put Cr- = lim{DIP" | n&N}. module.

The D - module CP» is called a Prufer P -

It follows from the construction of C/»» that Q/> j n (0°) =D DIP" and Q / \ n + i ( C » ) / Q ; > ( C » = (DIP"+l)l(DIP") = DIP. Hence if C is a proper D submodule of 0 ° , then C = Clp,„Cp™ for some n e N. Moreover, if c e Clpt„(Cp°>)\Qpt„-i(Cp«), then C = cD. A Prufer P-module is monolithic with monolith Qpj (C/>»).

47

48

5.1. Lemma

Just Infinite Modules

A Prtifer P - module is D - divisible.

Proof Let a e O » , 0 <£ x e D. There is a number n such that a e fi^(Cf)\Qw(Cf). It follows that a£> = Q P j „(C/»). Let / = xD. If IS P, then I+P = D. In this case D = 7 + P", so that 1 = xu+y for some u e £>,>> e P". Then a = a l = axu + ay = axu = (au)x. Consider the case when x e P. Put R = (") reN Ps • If we assume that R * < 0 >, then i? = Pi... Pt for some maximal ideals Pi,... ,Pt (see, for example, [KG, Theorem 3.3.5]). Since Pi... P, < Ps for any * e N, s < t (see, for example, [KG, Lemma 3.3.4]). So, we obtain a contradiction. This proves that R = < 0 >. Hence there exists a number t such thatx e P'\P'+1. Let B = Q/>in+;(C/>«>). The mapping

,„-i(CP«), then (Bx)P"-1 =< 0 >, that is AnnD(B) < xPn~l < P<+«-i. However, AnnD(B) = P"+l. This contradiction shows that //wq? = Q/>,„(C/>»). In particular, a e //wp, that is a = bx for some element b e B. 5.2. Proposition. Zef D be a Dedekind domain, P a maximal ideal of D, y € P\P2, A aP - module over the ring D. Then A is D- divisible if and only if A =Ay. Proof If A is divisible, then A = Ay. Conversely, assume that A = Ay. Let a e A,0 * x e D. If x £ P, then xD + P = D. Since ^ is a P - module, Annoict) — Ps for some s e N . It is easy to see that xD + Ps = D, so that 1 = xu+y where u e D,y e Ps. We have a = a l = a(;tw+j/) = axu + ay = az£t = (aw)x. Let now x e P. The equation A = Ay implies that A = Ay" for each n e N. We have xD = P'Q where P+Q = D (see, for example, [KG, Lemma 3.3.4]). It follows that x e P'\P'+1, so x = xiz where x\ e P', z e g . Moreover z £ P. Since ,4 = 4y', « = «iv' for some element a\ e ,4. Since Annoia) = Ps, we have Ps+I < AmtD(ai). Lemma 4.2 yields that P' = Dy' + P'+s, so x\ = y'u + v, u e D,v e /" +,s . From x i. P'+i it follows that u £ P. Since P is a prime ideal, u £ P, and z £ P, we can conclude that uz £ P. Hence uzP + P = D. This implies that uzD + P'+s = D, so 1 = uzx2 +*3 where *2 e D,X3 e P' + f . Then ai = ail = ai(uzx2 +X3) — (aiX2)uz + aixj = aiuz where 122 = aiXi. In particular, Anno{a{) < Anno{ai), that is P'+s < Annoiai). It follows that a = aiy' = (a2uz)y' = a2uzy' + a2vz = a2(uzy' + vz) = a2X, so A = Ax. 5.3. Proposition Let D be a Dedekind domain, P a maximal ideal of D,y e P\P2, A aP - module over the ring D. Then APS = Ays for every s e N. Proof Let a e AP. Then a = aixi + ... + a„x„ where a, e A,xt e P, 1 < / < n. Since A is a P - module, Annoixi) = Pkf for some A;, e N, 1 < i < n. Let

Some Results on Modules over Dedekind Domains

49

k = max{k\,... ,k„}; then ajPk = < 0 > for each i,I < i < n. Lemma 4.2 implies that P = yD + Pk, so thatx, = yut + v, where w, e Z),v, e Pk,l < i < n. We have a

= Ei<,<„«^' = Z,<,<„«'(V"- + v,) =

where c = ^ 1 < & a,«,. This means that a e 4y, that is AP = 4y. Let s > 1. We have already proved that APS = Ay". Put £ = APS. Then ^p,+l = (^/>^)p = BP = By = (4^)3, = ^ + 1 5.4. Corollary Let D be a Dedekind domain, P a maximal ideal ofD, A a P module over a ring D. If dimoipAlAP is finite, then A = B © C where B is a finitely generated submodule, C is a divisible submodule. Proof Let AIAP = ©, S £ n (a,D + ^(P), A0 = S ^ ^ f l / A then A = A0 +AP. Since Ao is a finitely generated P - module, there is a number si such that PSI < Anno(Ao). Therefore there exists a number £2 such that Ao yS2 = < 0 > but Aoysl~l * < 0 >. Put C = Ay*1, then C = Ay'* = (Ao +AP)yS2 = (A0 + Ay)ysi = A0ySl +Aysi+l = Aysi+l = (Aysi)y = Cy Proposition 5.3 implies that C is a divisible submodule, therefore A = C © B for some submodule B (see, for example,[SV, Proposition 2.10 and Theorem 2.15]). Furthermore, A=Ao+AP = A0 + (Ao +AP)P = Ao +AP2 = ... = A0 +APS> =A0 + C. Now B = A/C = (Ao + C)/C = Aol(Ao fl C) is a finitely generated submodule. 5.5. Corollary Let D be a Dedekind domain, P a maximal ideal of D, A a P module over the ring D. If A is an artinian D - module, then A = a\D © ... © asD © C\ © ... © C, where C, is a Priifer P- module, 1 < i < t. Proof We can consider AIAP as a vector space over the field DIP. Since A is artinian then dimD/PAIAP is finite. Corollary 5.4 implies that^4 = B © C where B is a finitely generated submodule, C is a divisible submodule. Then B = ai£> © ... © asD (see, for example, [CUR 1, Corollary 22.16). Let 0 * c\ e C, y e P\P 2 . Since C is D - divisible, there are elements {c„ | n e N}

50

Just Infinite Modules

with the following property: c„+\y = c„, n e N. It is not hard to prove that c„D =D DIP" for each n e N. It follows that C\ = Y^„m C"D i s a P r t i f e r submodule. In particular, C\ is D - divisible by Lemma 5.1. Now we can note that every D - divisible submodule is a direct summand of the module (see, for example, [SV, Proposition 2.10 and Theorem 2.15]). 5.6. Lemma Let D be a Dedekind domain, P a maximal ideal of D, A a P -module over the ring D. Then A is an artinian module if and only if dimD/pQp,i(A) is finite. Proof If A is artinian, then Q^i (A) is artinian too. It follows that dimD/PQ.p,\(A) is finite. Lemma 1 from [ZD 5] implies the converse. 5.7. Theorem Let D be a Dedekind domain, A a D - module. Then A is artinian if and only if A = a\D © ... © asD © C\ © ... © C,, where C, is a Prufer P, module, Annoicij) = PJ,Pj,Pj are maximal ideals ofD, 1 < i < t, 1 <j < s. Proof Suppose that A is artinian. If A is not D - periodic, then A has an element a such that Anno(a) =< 0 >. Therefore aD = D. However D has an infinite descending chain D > P > P2 > ... > Ps > ... for every P e Spec{D). This proves that A is D -periodic. By Proposition 1.1, A = © p e n o u ) Ap where Ap is the P - component of A. Since A is artinian, the set TID(A) is finite. Now we can apply Corollary 5.5 to each submodule Ap. The sufficiency follows from Lemma 5.6. 5.8. Lemma Let D be a Dedekind domain, P e Spec(D), A a torsion -free Dmodule and ro(A) = 1. IfB is a finitely generated non-zero submodule of A, then either the P - component of AIB is a cyclic P - module or it is a Prufer P module. Proof Let F be the field of fractions of the ring D, 0 * b e B. If bx = by for some elements x,y e D, then 0 = bx-by = b(x-y). Since A is torsion-free, x —y = 0, i.e. x = y. Therefore if we put (bx)

is a D - homomorphism and Im

i = (axi)'l ) = (bx2)<& = (bxi)<& = x2. It follows that ax\ = 0, so a = 0 because A is torsion - free. Hence <1> is a

Some Results on Modules over Dedekind Domains

51

monomorphism. Let RID be the P - component of FID, RXID = QPA(RID). If y e P\P2, then R\y < D. The mapping y : Ri —* D, defined by the rule xy/ = xy, x e Ri, is a D - homomorphism of Ri in D. Since F does not have a zero-divisor, Ken// = < 0 >, i.e. i?i = //wy = /. Moreover, / is an ideal of D. We have I/IP s Z)/P (see, for example the proof of Corollary 3.3.15 from [KG]). So and Ri/RiP = DIP. It follows that R\ID = DIP. Lemma 5.6 implies that RID is an artinian D- module, and Corollary 5.5 shows that RID = Cp°°. The factor-module AlbD is isomorphic with some submodule of FID, thus the P - component of AlbD is isomorphic with a submodule of Cp°>, i.e. either it is a cyclic submodule or it is isomorphic with 5.9. Corollary Let D be a Dedekind domain, P e Spec{D), A a torsion -free D - module, and ro(A) = 1. Then either there is a non-zero finitely generated submodule B such thatAlB is periodic andP £ HD{AIB), or the P -component of A/C is isomorphic with Cp» for every non-zero finitely generated submodule C. Proof. If for every finitely generated submodule C the P - component of AIC is isomorphic to 0 » , then the statement is proved. Therefore suppose that for some finitely generated submodule t/the P - component B/U of All] is not isomorphic with C/». By Lemma 5.8 this means that B/U is cyclic. It follows that B is finitely generated. Since BIU is the P - component of AIU, P £ YID(A/B). 5.10. Lemma

Let Dbea Dedekind domain, I an ideal ofD, AaD- module, < 0 > = Ao < A i < ... Aa < Aa+\ < ...Ar = A

an ascending series of submodules. If Aa+i/Aa = (Aa+\/Aa)Ifor any a 1 and assume that we have already proved the equation (Ap)I = A for all p < a. If a is a limit ordinal, then

If there exists a - 1 , then the submodule B = Aa-\ satisfies the condition B = BI. Let a e Aa. From (Aa/B) = (AJB)I ={AaI+ B)IB it follows that a = a\x\ + ... + a„x„ + b where ct\,... ,a„ e Aa,x\,...,x„ e I,b e B.

Just Infinite Modules

52 From B = BI it follows that

b = 6izi + ... + bizt where ft 1,... ,b, e B,zi,...,xt

e 7. Thus

a = a\X\ + ... + a„x„ + b\Z\ + ... + btzt e ^4 a / It follows thatv4 a = AaI. For a = 7 we obtain that ,4 = AI. 5.11.Lemma Let D be a Dedekind domain, P e Spec(D), A a torsion -free D module, and ro(A) = 1. Then either A = AP, or A includes a finitely generated submoduleBsuch that (AIB)P = AIBandB f]AP" = BP"for anyn&N Proof Assume that the P - component of AIC is a Priifer P - module for every non-zero finitely generated submodule C. Let 0 * a e A, RJaD be the P component of AlaD, 0 * x s P,a\ e R. Put a-i = a\x,C = c^^-Then RIC = Ui/C © UilC where U2/C is the P - component of RIC and U\IC is the P'component of RIC. Clearly U\IC is finitely generated and D - periodic. Since P £ UD{AIR), U2IC is the P - component of AIC. It follows that U2IC is a Priifer P - module, in particular, (t/2/Ox = U2IC because a Priifer P - module is divisible by Lemma 5.1. Let a\ + C = (u\ +C) + (1/2 + C) where u\ e U\,U2 e Ui- Then «2 = U3X + C1 where c\ e C. In turn, c\ = a2Zi for somezi e D, that is ci = {a\x)z\ = {a\z\)x. Hence «2 = U3X + c\ = U3X + (a\z\)x and m, e R. Since P £ YlD(UilQ, words,

= (U3 + a\Z\)x = U4X,

Lemma 1.3 yields that UylC = (U\IC)P.

In other

u\ + C = (vi + C)x\ + ... + (y, + C)x, for some elements vi,...,v, e U\,x\,...,x,

e P. Then

U\ = V\X\ + ... + VtXt + C2 where C2 e C. We can write C2 = «2Z2 = (<2i*)z2 = (a\Z2)x for some element Z3 e Z), so that u\ = v\X{ + ... + v,x, + {a\Z2)x e RP. Finally, a\ = u\ +U2+C3 where C3 e C, thus C3 = ci2Z3 = {a\x)z3 = (aiZ3)x for some element Z2 e Z). This means that «i e PP; hence R = RP.

Some Results on Modules over Dedekind Domains

53

Since AIR is D - periodic and P <£ UD(A/R), (AIR)P = AIR by Lemma 1,3. From the equalities R = RP, (AIR)P = AIR , and from Lemma 5.10 it follows that A = AP. Suppose that there is a finitely generated submodule B such that AIB is a D periodic module with the property P £ TLD(A/B). Put Bi = BP" Then BIB\ is the P - component of AIB\. Thus AIBX = BIB\ ®EIB\ where EIB\ is the P>component of AIB\. It follows that {AIB\)P" < EIB\. On the other hand, Lemma 1.3 implies that {EIB\)P" = EIB\. Consequently, E/Bi = EIBP" = {AlBP")Pn = {APn + BP")IBP" = AP"IBP" It follows t h a t ^ P " n B = E n B = BP". 5.12.Lemma Let D be a Dedekind domain, I a non-zero ideal of D, A a D module, B a submodule of A. If AIB is D - torsion -free then, B f\ AI = Bl. Proof Let b e BDAI. Since A = AIB is torsion - free, B = BIBI is its D periodic part. There exists a submodule C such that A = B © C [KTJ. Then AI = BI®CI= CISC. In particular, AlV\B= < 0 >, so b + BI= Bl. Thus BDAI=BI. Let D be a Dedekind domain, A a D - module, B a submodule of A, x e D. We say that B is anx - pure submodule ifBx = B n Ax. If I is an ideal ofD and B is an x - pure submodule of A for every x e /, then B is called an I - pure submodule. B is called a pure submodule if it is D - pure. 5.13.Lemma Let D be a Dedekind domain, x e D, A a D - module. IfTis a linearly ordered (by inclusion) set ofx - pure submodules, then the submodule B = \\X is x - pure too. Proof Let b e Ax fl B. Since b e B, b e C for some submodule C e I , that is b G Ax fl C. Since C is x - pure, Ax fl C = Cx, so that b e Cx. The inclusion Bx < Ax fl B is obvious. 5.14. Proposition Let D be a Dedekind domain, A a D - torsion-free D module, B a submodule of A. Then B is pure if and only if AIB is D - torsion -free. Proof Let B be a pure submodule of A, TIB a D - periodic part of AIB, a e T. Then ax e B for some element 0 * i e D , Since B fl Ax = Bx, ax = bx for some element b e B. It follows that (a - b)x = 0, i.e. x e Annoia - b). Since A is D torsion - free, a- b = 0, i.e. a = b. This means that T = B , i.e. AIB is £> - torsion - free.

54

Just Infinite Modules

Conversely, let A/B be D - torsion - free, ax e B for some x e D. If a £ B, then a + B is a non-zero D - periodic element of A/B. However, this is impossible. Hence a e B, i.e. AxC\B = Bx and B is a pure submodule. 5.15. Proposition Let D be a Dedekind domain, A a torsion -free D - module. Then A has an ascending series of pure submodules < 0 > = Ao < A\ < ...Aa < Aa+i < ...Ar = A such that rD{Aa+\IAa) = 1 for each a 1 and suppose that we have already constructed the submodules Ap for all B < a. If a is a limit ordinal then put Aa = [Ja C. IfCC\AI= CI, (A/QI n (BIQ = (B/QI, then AIHB = BI. Proof Let b e BC\AI. Then b + C e (AI+C/Q n (B/Q = (A/QI n(B/Q = (B/C)I = BI+C/C, that is b = b\X\+ ... + b„x„ +c for some b\,... ,b„ e B, c e C,xi,... ,x„ e I. It follows that c = b- b\X\ - ... - bnxn e AI, so c e AID C = CI. Hence ft e 67+ CI < BI, i.e. 5 f]AI < BI. 5.17.Corollary Let D be a Dedekind domain, I an ideal ofD, A a D - module, B a submodule having an ascending series of submodules < 0 >= Bo < Bx < ... Ba < Ba+i < ... By = B. If(Ba+l/Ba)

n (AIBa)I = (Ba+i/Ba)Ifor

every a < y, then BnAI

= BI.

Proof We will use induction on a. If a = 1, then the assertion is true. Assume thata > 1 and we have already proved that AI n Bp = (Bp)I for every R < a.If a

Some Results on Modules over Dedekind Domains

55

is a limit ordinal, then Ba = lL< a &P> s o AlnBa=Aln\JP
= {j^AinBp)

= \Jp
If there exists a-I, then Ba-\ C\AI = Ba-\I hy the induction hypothesis. Lemma 5.16 implies that BaC\AI = BaI. For a = y we obtain the equation BI = B fl ^4/. £e£ D be a Dedekind domain, P e Spec(D), AaD -module. A submodule B is called aP - basic submodule if it satisfies the following conditions : (1) B = B\ © #2 w/jere Bi is a direct sum of cyclic P - submodules, Bi is a projective D - submodule; (2) (A/B)P = AIB; (3) 5 n AP" = BP"/or a«y n e N. 5.18. Theorem Z,e/ D be a Dedekind domain, P e Spec{D), A a torsion -free D - module. Then A has a P - basic submodule. Proof

Proposition 5.15 implies that A has an ascending series of submodules < 0 > = Ao
such OaaXAIAa is D - torsion - free and rD(Aa+\/Aa) = 1 for every a < y. We will apply induction on a. Moreover, we will prove that Aa includes a submodule Ba satisfying the following conditions: Ba r\Aa-i = Ba-\ (in particular, Ba-i < Ba)\ Ba and BJBa-i are projective modules; {AJBa)P = AJBa; Ba f\AaP" = BaP" for any n e N. If a = 1, then we can use Lemma 5.11. Let a > 1 and we have already constructed the submodules Bp for all B
= AplBp,

in particular, ((Ap+Ba)/Ba)P

= (Ap+BayBa

56

Just Infinite Modules

for every B < a. Finally, Corollary 5.17 implies that BanAaPn

=BaP„.

Now assume that a - 1 exists. Then Aa-\ includes a projective D - submodule B„_i such that (Aa-,/Ba-i)P = Aa-\IBa-\ and Ba-iP" = Ba-\ f\Aa-\P" for each n e N. From Lemma 5.11, it follows that Aa/Aa-\ includes a finitely generated submodule CA4„_i such that (AJQP = AJC, (CIAa-\) f]{AaIAa-i)P" = (C/Aa-i)P" for each n e N. Hence C/Ba-i = (Aa-i/Ba-i) © (BaIBa-i) and Ba is a projective submodule (see, for example, [PD 2, Lemma 6.1]). Using the isomorphism C/B„ = Aa-\IBa-\ we obtain that (CIBa)P = C/Ba. From the equality (AJQP = AJC and Lemma 5.10 it follows that (AJBa)P = AJBa. Finally, assuming that b e Ba f]AaP", we have b +Aa-i e (AaP" +Aa-i/Aa-i) n (Ba +Aa-ilAa-i) = (AaIAa-i)P" D (CIAa-X) = (CIAa-{)P"

= (Ba + Aa-1IAa-l)P"

=

(BaPn+Aa-O/Aa-U

In other words, b = b\xi + ... + bsxs + e where b\,... ,bs e

5„,JCI,...

,xs e /"", e e Aa-\. Furthermore, e = i - i i ^ i - ... - bsxs,

that is e e B„ D^a-i = 5 a -i, thus e e 5 a -i C\AaP". Since AaIAa-\ is D - torsion - free, Lemma 5.12 yields thaU a -i (~\AaP" = Aa-XP". So e e B a _j f\AaP"

= 5 a _ , n ^ a - l f l ^ a P " = Ba-y HAa-lP"

=

Ba-XP".

It follows that b e BaP". 5.19. Lemma le/ D be a Dedekind domain, P e Spec(D), A a D - module. Suppose that A is D - torsion - free and ro(A) is finite. Let B be a finitely generated submodule of A such that AlB is D - periodic. If YIB is the P component ofAlB, then YIB = Ci © ... © Ck © Ex © - © En where C, is a Prtifer P - module, I < i < k,Ej is a cyclic submodule, 1

<j
Some Results on Modules over Dedekind Domains

57 2

Proof Let YXIB = Q/>,i(7/B). Then YXP < B. Let z e P\P mapping

Consider the

q>z : a —• az, a e Y\. Clearly Imq>z < B. Since A is D - torsion - free, Ker, in particular, Y\ = Imq>z. Hence Y\IB = (]mfz)/{B», 1 < ;' < k,Ej is a cyclic submodule, 1

<j
5.20. Lemma Le* D be a Dedekind domain, P e Spec(D), A a D - module, B a submodule ofA such that AIB is D - torsion -free, and (A/B)P = AIB. Then A/AP" = B/BP" for every n e N. Proof Since AIB is torsion - free then B = BIBP" is the D - periodic part of A = AIBP". There is a D - submodule C such that .4 = B © C [KI], Moreover, C = ^75 = (AIBP")l(BIBPn) £ ^ / 5 , in particular, CP" = C. It follows that AP" = fiF" © CP" = C. Hence ^/^IP" = (A/BP")/((A/AP")/(BPn)) = ^/C = (5 © C)/C £ B = B/BP". 5.21. Lemma Let Dbe a Dedekind domain, P e Spec{D),A a torsion -free D module of finite D - rank, B a finitely generated submodule ofA such that AIB is D - periodic. IfB is aP - basic submodule ofA, then AIAP" = BIBP" for every n e N. Proof Let YIB be the P - component of AIB. Proposition 1.1 yields that AIB = YIB © QIB, where QIB is the PI- component of AIB. In particular, (Q/B)P = Q/B by Lemma 1.3. Thus {AIB)P = {YIB)P © {QIB). Since {AIB)P = AIB, {YIB)P = YIB. Lemma 5.19 yields that YIB is a direct sum of finitely many Priifer P - submodules. Lemma 5.1 implies that YIB is divisible. Consider the factor - module AIBP". Clearly, Y/BP" is the P - component of AIBP". Lemma 5.19 proves that Y/BP" = EIBP" © CIBP" where E/BP" is the D -

58

Just Infinite Modules

divisible part of YIBP", CIBP" is a finitely generated submodule. Since BIBP" is finitely generated, (E/BP" + B/BP")I(B/BP") = ((E + B)/BP")/(B/BP") == (E + B)/B is the D - divisible part of YIB. But YIB is D - divisible, that is (E + B)IB = Y/B._ Put now A = AIBP",B = B/BP",E = E/BP", Y = YIBP". Hence A = Y® Qx where Q\ is the Pi- component of A. We have now AIAP" = {AIBP")l{AIBP")Pn = (A/BP")/((AP" + BP")IBP") = (AIBP")I{AP"IBP") = AIAP". Lemma 1.3 implies that QXP" = Qu so that AP" = YP" © g i , that is AIAP" = {Y® Qx)l(YP" ®Q\)= YIYP". Finally, Y = B + E, therefore YP" = E, thus YIYP" = (B + E)IE = 5/(5 n £) = (B/BP")I(B/BP" f] E/BP") = {BIBP")I{{B n E)IBP"). Since 5 is a finitely generated torsion - free module, 5 = (£> ® ... ® Z)) ©7 r-l

where 7 is some ideal of D (see, for example, [CUR 1, Theorem 4.13] and [PD 2, Theorem 7.7]). Since HIP" = D/P", it follows that there is a sum of r summands BIBP" =D/P" ®...®D/P". r

Hence if we assume that 5 fl E * 5P", then 5/(5 f] E) has no submodules which is isomorphic with BIBP". On the other hand, since 5 is a P - basic submodule, 5 n AP" = BP". So that BIBP" = Bl(BnAP") s {B + AP")IAP", in particular, AIAP" (and hence YIYP") includes a submodule, which is isomorphic to BIBP". This means that Bf]E = BP" so that P/IT" = BIBP". Thus y4A4P" = i//fP" = YIYP" = BIBP". 5.22. Proposition

Let D be a Dedekind domain, P e Spec(D), A a torsion -

Some Results on Modules over Dedekind Domains

59

free D - module, B aP - basic submodule ofA. lfB has finite D - rank, then AIAP" £ BIBP" for every n e N. Proof Let TIB be the D - periodic part ofAIB. Since B is a P - basis submodule, (A/B)P = A/B. Therefore and {AIT)P = AIT. Lemma 5.20 yields that AIAP" = T/TP" for any n e N. Since B is a projective submodule of finite D - rank, B is finitely generated. This means that roCT) is finite. Since AIT is torsion - free, Lemma 5.12 proves that TP" = TC\AP", thus we have BHTP" = BDTDAP"

= BDAP" = BP".

Suppose that {TIB)P * 77B. Let A = (AIB)I(TIB)P, T = (T/B)I{T/B)P. Since f is the D - jjeriodic part of ^4 and P"_< AnnoCT), A = T © C for some D submodule C [KI]. From the election of B it follows that AP = /f. However, AP =TP®CP

= CP * X

This contradiction shows that (T/B)P = TAB. In other words, £ is a P - basic submodule of T. Lemma 5.21 proves that in this case T/(TP") == B/(BP"), and therefore A/(AP") = T/(TP") = BI{BP"). 5.23. Corollary Le/ D be a Dedekind domain, P e Spec{D), A a torsion -free D - module. Ifdimoip{AIAP) = m is finite, then AIAP" is a sum ofm summands AIAP" = DIP" © ... © DIP" for any n e N. Proof Let B be a P - basic submodule of ,4. Since Bf]AP = BP, BIBP = B/(AP n B) s (B+AP)/AP, so that dimD/P(B/BP) is finite. Assume that £ is not finitely generated. Since B is a projective module, B is a free D - module (see, for example [PD 2, Theorem 7.7]). As a free module of infinite D - rank, the factor module BIBP is infinite dimensional. Thus B is finitely generated, and we can use Proposition 5.22. OLet A be a module over a ring R, B a submodule ofA. We say that B has a complement in A {or B is a complemented submodule) if there is an R - submodule C such that A = B © C.

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Chapter 6 Just Infinite Modules over FC-Hypercentral Groups

Now, we are ready to start our study. Our first step is Theorem 6.6, allowing us "to change" the underlying ring of coefficients. The main part of this chapter is devoted to applications of this theorem to just infinite modules over FC hypercentral groups. All these applications are based on reduction to a normal subgroup of finite index. Let D be a Dedekind domain and G a group. Let A be a non-zero just infinite DG - module. By definition (see (JI 2)), there exists a non-zero proper DG submodule B. In particular, AIB is finite, so that there exists a maximal DG submodule M such that M>B. If D is finite, then it turns out that D is a (finite) field. If D is infinite, then Ann^iAIM) = P is a maximal ideal of D. Since AIM is finite, it follows that F = DIP is a finite field. This simple remark makes sense to the following definition. An infinite Dedekind domain D is said to be a Dedekind ZQ - domain, if for every maximal ideal P ofD the factor - ring DIP is finite. 6.1. Lemma Let A be a just infinite module over a ring R,

is a monomorphism. Proof Since q> is a non-zero R - endomorphism, Ker

, then AIKer

, that is

. 6.2. Corollary

Let A be a just infinite module over a ring R. Then EndR(A) has

61

62

Just Infinite Modules

no zero-divisors. 6.3. Corollary Let D be a Dedekind Zo - domain, G a group, A a just infinite module over a group ring DG, I - AnnoG{A), CII the center of the factor-ring DGII. Then CII is an integral domain. Proof For each element x e C the mapping ix : a —• ax, a e A, is a DG endomorphism of A. Furthermore, the mapping : x —• ix, x e C, obviously, is a homomorphism of the ring C in the ring EndoG(A) and Ker or (D 0 AnnoG(A)) is a maximal ideal of D. 6.5. Corollary Let D be a Dedekind Zo - domain, G a group, U z e C,(G),A a just infinite DG - module, CG(A) = < 1 >. Then CA(Z) = < 0 >. Proof

Since z e C, (G), the mapping

is a DG - endomorphism of A. By Lemma 6.1 Ker

. 6.6. Theorem Let D be a Dedekind Zo - domain, G a group, A a just infinite DG - module, which is D - torsion -free, CG(A) = < 1 >. Then D is embedded in a principal ideal domain J and the DG - module A is embedded in a JG - module Vwith the following properties: (1) Vis a free J- module; (2) V isfinitelygenerated as J - module; (3)CG(F)=<1>; (4) V is a just infinite JG - module. Proof If Q is a proper non-zero DG- submodule of A (such submodule exists by (JI 2)), then AIQ is a finite D - module. It follows that AnnD{AIQ) * < 0 >. Hence there exists an ideal P e Spec(D) such that A * AP. Let A/AP = Bn®...®Bri where Bn = DIP, \
Just Infinite Modules over FC-Hypercentral Groups

AIAP" = B\„ © ... © Brn where Bin = DIP",

63

\
Let f„m : v4/^P" — ,4A4Pm, m < n, be a canonical epimorphism. Then {AIAP",(pnm

\m
is a projective family of finite D - modules. Put V = lim{A/AP",(pnm \m
m,n e N>.

Then Fis a .D - module and since < 0 > = P)„eN ^4^", the mapping a ->• ( a + ^ P " ) „ e N , f l e A,

is an embedding of A in F by Remak's theorem. Indeed, by Corollary 5.23 AP" * APn+l for any n e N. It follows that f\6N ^ " has an infinite index in A. By (JI 2) p) n e N ^ P " = < 0 >. In the sequel, we will identify A with its image in V. Clearly, we can choose the submodules Bin such that Bin(p„m = Bim, \ < i < r, m < n. Elementary properties of projective limits imply that F = W\ © ... © Wr, where Wj = lim{Bin,(pnm | m < n, m,n e N}. Obviously, W,• = J = lim{DlPn \m
N}

We can define an action of the group G on Fas follows. Let (a„)„eN e F; then a„ = a„ +AP" for suitable a„ e ^4, and a„
Just Infinite Modules

64

(a„g), we have CG(V) = < 1 > . The action of D on F could be naturally extended to an action of Jon V. Now, VP" = lim{A/AP"+' | f € N} = V„ (see, for example, [ND, 9.10 Theorem 18]), and VIVP" = A/AP", n e N (see, for example, [ND, 9.5, Corollary of Theorem 4]). Let.y e P\P2; then P = Dy + P" for any « e N, in particular, Pm = Dym + P" if w < « by Lemma 4.2. Let (x„)„s^ e U(J); then there is an element (U„)„<EN e J such that (x„)neN(wn)«eN = 1 where 1 = (1 + P,\+P2,... ,\+P",...). In particular, 5c\u\ = 1 + P, so that it] * 0. Conversely, let (3C„)„SN e J and ici * 0. Then 3c i = x\ + P * />, that is x\ £ P. Since P is a maximal ideal of D then P + x\D = D. Let 3c„ = JC + P"; then x„ + Pm = xm + Pm for w < n, in particular, x„ i P for any w e N. Again P + x„D = Z) and therefore x„D + P" = D. Hence there are elements u„ s D,v„ e P" such that JC„M„+VB = 1, or 1 + P" = x„u„ + v„ + P" = x„w„ + />" = (x„ + />")("« + P") = x„u„, where tin = un+P". Since the inverse element is unique, u„ + Pm = um + Pm for m < n, so (M„)„£N e J. Hence U(J) = {(xn)„GN\xi

*0>.

Let 6 =_(Z>„)„<=N bea non-zero element of V, we set ? to be the number with the property b\ = ... = b, = 0 but S,+i * 0. Note that VP' = Vy'. Indeed, let Cs«)«eN e J, x e / " . From the equation / " = Z)y' + P'+m it follows that x = y'zun + ~wm where z,m e £>, w,m e 7>'+m. Then {{sn)nm)x = (s„x)„eN = (s„(y'z,m +w,m))„m = (0,... ,0,St+l(y'z,i

+W,l),...

,~St+m(y'Ztm + Wtm),..)

=

(0,... ,Q,s,+\y'z,\,... ,'s,+my'ztm,...) = (0,... ,0,5,+iz,i,... ,s,+mz,m,...)y' since wtm e F' +m = AnnD(DIP'+m),m € N. From the equation J/" = j y we obtain that b = cy' where c e U(J). The D module J is torsion-free (see, for example, [ND, 9.10, Proposition 14]), so the above expression is unique. If rf is an other non-zero element of J, then again d = eym where e e U(J). Thus W = cey'+m * O.SoJ is an integral domain. Let I be a non-zero ideal of J. For given 0 * b e I, we can write again 6 = cy'

Just Infinite Modules over FC-Hypercentral Groups

65 +l

where c e U(J). It follows that Jy' < I. However, Jy' = JP' = Um{DIP' \ I G N>, and J/JP' m DIP'. The set {DIP', PIP',... ,P'-lIP',< 0 >} is the set of all ideals of DIP' by Lemma 4.2. Hence 7/JP' is isomorphic to P'~kIP' for some k. In other words, I = JP'"* = Jy' - *. This means that J is a principal ideal domain and every ideal of J has finite index. Let U be a non-zero J - submodule of V, U\ a J - pure envelope of U (i.e. U\IU is the J - periodic part of VIU). Since J - rank of V is r then rj(Ui) = rj(V) = q < r. We claim that U\IU is really finite. To show this, we note that VIU is a finitely generated J-module. Since J is a principal ideal domain, it is a noetherian ring. In particular, Ui/U is finitely generated. Now, we may observe that every finitely generated J- periodic module is finite, since every non-zero ideal of J has finite index. Now we want to prove that U\ f]A * < 0 >. Since \A/AP"\ = \VIVP"\, we can write that .4 + VP" = Kand.4 |~l VP" = AP", n e N. Therefore, VIU i = (A + Ui)/Ui +(VP" + Ui)/Uu and ((A + U\)IUX) n {{VPn + U\)IU\) = {AP" + Ui)/Ui. From VIUiP" = (VP" + U\)IU\ it follows that (V/Ui)/(V/Ui)P" = ({A + Ui)/Ui)) +{VP" + U\)IUX)I(VP" + U\)IUi) = ((A + Uy)IUX)l{{A + UX)IUy) fl {VP" + Ui)IUx)) = {{A + Ui)/Uiy{{AP" + Ul)/Ui) = {{A + Ux)IUx)l{{A + Ui)/Ui)P") s (4/(4 n Ui))/(A/(A f) Ui))P«. Since V = U\ © M, we have (VIU\)I(VIUX)P"

= M/MP", and MIMP" = DIP" © ... © DIP" . r-q

If we assume that A f l t / i = < 0 >, then (A/(A n Ui))/(A/(A n f/i))/"" = ^ / ^ / J " = DIP" © . © £>/P". r

This contradiction shows that^ fl U\ * < 0 >. Let t/ be a non-zero JG - submodule of V, U\ be a J - pure envelope of U in K Clearly U\ is a JG - submodule of K. As we have already shown, A(~\U\ is a non-zero DG - submodule of A, so the index \A : (A fl Ui)\ is finite. However, VIUi is J - torsion-free, so it follows that ,4 C\U\ = A.

Just Infinite Modules

66

Let a, be an element of A such that Bt\ = a\D + AP, 1 < / < r. Then V = a\J@ ... ®arJ. Since au... ,ar e Uu V=Uy. Thus \V: U\ = \U\ : U\ Consequently, every non-zero JG - submodule of Fhas finite index. Finally, the equation

is

finite.

VP" = lim{A/AP"+I | / e N} implies

r u ">»=<<>>. Hence J7 satisfies (JI 2). Thus Fis a just infinite JG - module. 6.7. Corollary Let D be a Dedekind Zo - domain, G a group, A a DG - module which is D - torsion-free, CQ(A) = < 1 >. If A is a just infinite DG - module, then there exists a field F > D and a simple FG - module B > A such that CQ(B) = < 1 >, and dim FB is finite. Proof Apply the above result. Let F be the field of fractions of the principal ideal domain J. On the other hand, B = V®j F. If E is a non-zero TO-submodule of B, then V C\ E is a non-zero JG-submodule of V. Hence the index | V : V C\ E\ is finite. In particular, the J - module B/(E C\ V) is periodic, so that and BIE is J periodic. From this, we must have E = B. In other words, B is a simple FG module. Since CG(V) = < 1 >, CG(B) = < 1 >. Finally, dimFB = rj{V), thus dimfB is finite. 6.8. Corollary Let D be a Dedekind Zo - domain, G a locally radical group, A a DG - module which is D - torsion-free, CG(A) = < 1 >. If A is a just infinite DG - module then G is abelian-by-finite. Proof Indeed, by Corollary 6.7 there are a field F > J and a simple FG- module B > V > A such that CG(B) = < 1 > and r = dimFB is finite. Under these conditions we can consider G as an irreducible subgroup of the linear group GLr(F). Since G is locally radical, then G is soluble (see, for example, [WB , Corollary 3.8]), therefore G is abelian-by-finite (see, for example, [WB , Lemma 3.5]). The statements 6.6 - 6.8 slightly generalize the main result of [KK]. The following result shows that in the study of just infinite modules it is

Just Infinite Modules over FC-Hypercenlral Groups

67

possible to apply the reduction to normal subgroups of finite index . 6.9. Proposition [FdeGK 3] Let R be a ring, G a group, A a just infinite RG module, CQ(A) = < 1 >. IfH is a normal subgroup of finite index ofG and X is a transversal to H in G, then A includes an RH - submodule B such that (i) AlBx is a just infinite RH - module for every x e X; (ii) A is isomorphic with an RH - submodule o / ® x e A . (A/Bx); (Hi) H is isomorphic to a subgroup ofXXsx(Hlx~lCx) Proof

where C =

CH(AIB).

Since A is a just infinite RG - module, it is noetherian. Put M = {Q\Q is an RH - submodule of A such that A/Q is infinite}.

Clearly M * 0. By [WJ 1, Theorem A] A is a noetherian RH - module. Hence the family M has a maximal element B. Obviously (~\xsX Bx is an RG - submodule of infinite index, therefore \^\xeX Bx = < 0 >. Assume that the RH - module AIB is not just infinite, and let AQ/B be the intersection of all non-zero RH - submodules of AIB. Then Ao * B, so A/Ao is finite. It follows that the RG- submodule A\ = |~|xeA- AQX has finite index in A. The RH - module AQIB is infinite simple, so AQX/BX is an infinite simple RH module for every x e X. Hence A\ + Bx = AQX for all x e X. By Remak's theorem, from the equation f*\xsX Bx =< 0 >we obtain the embedding A\ < © ^ 1 / ( ^ 1 l~l fix). Furthermore, A\I{A\ C\Bx) £ (A\ + Bx)IBx = Aox/Bx is a simple RH module, in particular, A i is a semisimple RH - module. Since every RH - module A\I{A\ fl Bx) is infinite, then A i includes an infinite simple RH - submodule E. If S is a non-zero RG- submodule of A, then AIS is finite, so EI(E fl S) is finite too. Thus E = E f| S, since E fl S is an RH - submodule of E, that is E < S. Hence the intersection of all non-zero RG - submodules of A includes E. However, this intersection is zero. This contradiction shows that Ao = B. In other words, AIB is a just infinite RH - module. It follows that AlBx is a just infinite RH - module for each x e X. Once more, from Remak's theorem we obtain the embedding A <

®x^AIBx.

Finally, CH(AIBx) = x'lCH(AIB)x for every x e X, and f\x£XCH(A/Bx) CH(A)=< 1 >, so H<XxeXH/(CH(A/BY).

=

Thus, if H is a normal subgroup of G and GIH is finite, then the structure of just infinite /W-modules provides a lot of information about just infinite WG-modules.

68

Just Infinite Modules

In a similar way, the structure of the automorphism group HICH(AIB) of AIB allows us to obtain information about G/CG(A). AS the following result shows, there is a dual connection. 6.10. Proposition [RW] Let R be a ring, G a group and H a normal subgroup ofG with finite GIH. If there exists a just infinite RH-module B such that CH{B) = < 1 >, then there exists a just infinite RG-module A such that CG(A) = < 1 >. Proof We start with transitive group F of permutations of a finite set Y. Given H and B, form the split extension Q = B x H and consider the wreath product E = Q wrF. Let D be its base group. Then D = Xy^rQy, where Qy = Q, y e Y. For any y e Y, let By the isomorphic image of B in Qy, and define A = Xy^yBy. Let xg e CE(A) where x e D, g e F. Since g centralizes the diagonal subgroup of A, then x also centralizes this subgroup. Further, x is really a function from Y to g, and therefore, given y e Y, x(y)b = bx(y), for every b <= B. In particular, x(y) e Cg(5) = B, for each^ e Y, which gives x e i . g e Q M and so g = 1. This means that CE(A) = A. Now let Y be the set of all cosets Y = {Hx \ x e G> of G modulo //. Then the group F = G/// acts transitively on Y and we may apply the above construction. We have

E\ (HwrF)

=A\(P\F),

where P is the base group of the wreath product H wr F. By [HUB, Theorem 1.15.9] G can be embedded in Hwr F, moreover, GP = Hwr F. In particular, for each element g e F, there exist elements hg e P and cg e G, such that g = /i g c g . Let £/ be an 7?G-submodule of A. Suppose that, for some y e Y, we have that f/n^j, = < 1 >. Then Uf] Qy = < 1 >, and it follows that [U,Qy] = < 1 >. Since F acts transitively on Y, it follows that cg = (hg)~xg acts on the direct factors of D in the same way as the element g does. This yields that [U,D] = < 1 >. However, it is easy to see that CE(D) = < 1 >, SO that the above is a contradiction. Consequently, for every y e Y, we have that Uf)By ±< 1 >. Then each By/(Uf)By) is finite, so A/(XyeY(U0 By) is finite too. From this, it follows that AIU is finite. Since B is a just infinite RH-moAu\e, the intersection of all non-zero ^G-submoduIes of A is zero. These assertions assure that A is a just infinite i?G-module, as required. To obtain more information on these just infinite modules we shall need some

Just Infinite Modules over FC-Hypercentral Groups

69

structural facts about linear groups. They will be exposed in a slightly general form. 6.11. Proposition [KO 2] Let Xbe aformation ofgroups, F a field, G < GL„{F). IfG is an XC - group, then Glt^{G) e X. Proof Consider G as a subset of the finite-dimensional F - space M„{F) of all n x n matrices over F. Let U be its F - subspace generated by G. Since dimpMniJF) = n2, dimpU is finite. We can consider U as an FG - submodule, defining a G-action on U by conjugation ug = g~xug,u € U,g e G. Let {e\,... ,er} be a basis of the subspace £/. Then e« = a/ig,i + ... + a„,g,;„ a,y e F, gtj e G,\ <j < th 1 < i < r. It follows that Cc(ep)>nisSjCG(gG),l
GICG{^)

e <%", because A" is formation. By the same

G/(CG(ef) n ... 0 C G (e?) g * Since G e [/,we can write C G (ef) l"l... fl C G (e°) < C(/(G) and G/£(G) e X, as required. Let Xbe a class of groups. Put S„ X= { G | G is a normal subgroup of some group L € X}, PX= { G | G has a finite subnormal series, every factor of which belongs to X }• 6.12. Proposition [KO 2] Let X be a formation of groups satisfying the following conditions: X = S„ X, X = PX, F afield, G < GL„(F). IfG is XC hypercentral, then G/Fitt(G) e X. Proof Let Vbe an F - subspace of M„(F), generated by G, and think of Vas an FG-module trough the conjugation by elements of G. Then dimpV= r < n2. Using induction on r we are going to show that there exist a basis of V and a

70

Ji4St Infinite Modules

subnormal series H = HK

H2 < ... < H, = G

such that every element of H has an unitriangular form in this basis, and Hi+i/Hj e Xfor every i, 1 < i < t - 1. Let A i = XC(G) ,V\ be a F - subspace of V generated by A i. Since A\ is normal in X, V\ is an FG - submodule. Put C\ = CG(V\). Using the same arguments as in Proposition 6.11, we can prove that C\ = CQ{A\) and GIC\ e X. Let {e\,... ,er\} be an F - basis of V\. Complete this basis to a basis of all space V. In this basis every element of Ci has the form

f

\_

\

0

...

0

0

1

...

0

«r,+ll

a n +12

ar\

ar2

A

— On+lr

...

arr

J

Fix c e C\ and define the mapping 0C : VI V\ —* VI V\ by the rule {v+Vi)0c = vc+Vuv

G V.

Clearly, 9C is a non-singular linear transformation of the F - space V/V\, and the mapping 0C, c e C\,

is a group homomorphism of Ci in GLr~ri (F). Since A" is closed under taking normal subgroups, then C\ is an XC - hypercentral group. By the induction hypothesis VIV\ has a basis {en+\ + V\,...,er+ V\} and C\

= Ci
such that every element of H has in this basis an unitriangular form and Hi+\IHj e X, 1 < i < t-2. Then {e\,... ,e„,e r i + i,... ,er> is a basis of V. For each j let Ht be the preimage of//, in C\ Then Ci has a subnormal series

Just Infinite Modules over FC-Hypercentral Groups

H=Hi
71

= C\

such that Hi+\IHi E X. Clearly, every element of H admits an unitriangular form with respect to this basis. Since H is an unitriangular subgroup, it is well-known that it is nilpotent. In particular, H\ < Fitt{H2), so HilFitt(Hi) e Xsince -Vis a formation. Recall that the Fitting subgroup of a linear group is nilpotent (see, for example, [WB, Theorem 8.2]), then H3 includes a normal nilpotent subgroup Fitt^Hj) such that H3/Fitt(H2) e PX= X. Proceeding in this way, we come to GIFitt{G) e X. The above proof has applied arguments from [MM], although we stated a more general result. This allows to us to apply Proposition 6.12 to several particular cases, namely for S„- and P-closed formations. Here are some examples. 6.13. Corollary [KO 2] Let X be a formation of groups, F afield, G an XC hypercentral subgroup ofGL„{F). {Y^IfX^F, then G is nilpotent-by-finite. (2) IfX- C, then G is nilpotent-by-Chernikov group. (3) If X = VT is a class of polycyclic-by-finite groups, then G is nilpotent-by-polycyclic-by-finite group. (4) If X = S3 J- is a class of soluble-by-finite minimax groups, then G is nilpotent-by-minimax and soluble-by-finite. Indeed, every class T, C, VT, S3J- are S„ - closed and P - closed, and the Fitting subgroup of a linear group is nilpotent (see, for example, [WB, Theorem 8.2]). 6.14. Lemma Let D be a Dedekind domain, / an ideal of D, G a group, A a DG - module. If x e CG(AIAI), then* e CG{AI"IAIn+x) for every « e N. Proof We will use induction on n. Suppose that we have already proved that x s CG(AI"~1/AI") for some n. Let a e AI". There are elements u\,U2,v\,V2 e D such that / = V1D + V2DJ" = u\D + ujD (see, for example, [KG, Corollary 3.3.14]). We have a = a\u\+ajU2 for some elements a\,a2 e A. Since at(x - 1) € AI, we can write ai(x- 1) = anv\ +ai2V2 for some elements a,y, i,j =1,2. Now we have a{x- 1) = (a\U\ +a2U2)(x- 1) = a\(x- l)m + a2{x- 1)«2 = (an vi + a\2v2)u\ + (a2ivi + a22V2)u2 = an vi«i + a\2v2u\ + a2\v\u2 + a22v2u2.

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Since v,w/ e I"+\ i,j e {1,2}, a(x - 1) e AI"+l, that isx e C G (^/"/^/" + 1 ). 6.15. Theorem [FdeGK 3] Let D be a Dedekind Z0 - domain, G an FC hypercentral group, A a just infinite DG - module which is D - torsion-free. If Cc(A) = < 1 >, then G includes a normal torsion-free abelian subgroup of finite index. Moreover, ifcharD = p > 0, then Op(G) = < 1 >. Proof Corollary 6.7 yields that G is imbedded in the group GL„(F) where F is a field including a ring D. Corollary 6.13 shows that G is nilpotent-by-finite. Let H be a normal nilpotent subgroup of G such that GIH is finite, X be a transversal to X in G. Proposition 6.9 yields that ^ includes a Z)// - submodule 5 such that A/Bx is a just infinite DH - module for every x e X. Corollary 6.8 proves that HICH{AIBx) is abelian-by-finite. By Proposition 6.9 H < (&x^HICH(AIBx), thus H, and hence G, is abelian-by-finite. Choose in G a normal abelian subgroup U of finite index. Let Q be a maximal ideal of D such that A * ^4g, q» = char(DIQ), C = Cu(A/AQ), c an element of C such that \c\ = k is finite and (&,<7) = 1. If a s A, then ac = a + a\ for suitable element a\ e AQ. LetA=A/AQ2,Ai = AQ/AQ2. Lemma 6.14 yields that c e C u ( ^ 2 ^ 2 2 ) - L e t a = a + , 4 g 2 , a i = a\ + ^ g 2 . T h e n ac" = a + na\, n e N, in particular, a = ack = a + M\. It follows that fan e , 4 0 2 . Since AIAQ2 is a g group and (&,g) = 1, this means that a\ e AQ2. In other words, c e CG(A/AQ2). Similarly we can prove that c e CQ(A/AQ") for n e N. By Corollary 5.23 ^ g " * ^ g ' * 1 ; hence f\„^AQn = < 0 >, and a(c - 1) = 0, so c e CG{A) =< 1 >. Hence the subgroup C cannot contain the q>- elements. Furthermore, the equation |~)neN^2" = < 0 > shows that HneN CG(AIAQ") < CG(A) = < 1 > . Together with Corollary 5.23 this implies that G is residually finite. Suppose that charD = 0; then charF = 0. Since G < GL„(F), all Sylow q subgroups of G are Chernikov for any prime q (see, for example, [WB, Theorem 9.1]). Moreover, they are finite (G is residually finite). We have already proved that Cu{AIAQ) does not contain the ql- elements. Since UICu(AIAQ) is finite, this means that the periodic part T of the abelian subgroup U is finite. By the residual finitenes of G it follows that U includes a G - invariant torsion-free subgroup of finite index (G is residually finite). Finally, let char D = p > 0. Similarly we can obtain that U includes a G invariant torsion-free subgroup of finite index. Suppose that Op(G) * < 1 >. Since G is an FC - hypercentral group, Corollary 3.4 yields that Op(G) fl FC(G) * < 1 >. Hence Op(G) includes a non-identity finite abelian G invariant subgroup E. Put W= CG(E); then W is a normal subgroup of finite

Just Infinite Modules oxer FC-Hypercentral Groups

73

index. Proposition 6.9 shows that A includes a DW - submodule B\ such that^4/5i is a just infinite DW- module. Since AIB\ is an elementary abelian p - group, it is easy to see that CA/B,(.E) * < 0 >. Since E is abelian, E < C,{W). Corollary 6.5 yields that in this case E < CG(AIB\). Let S be a transversal to W in G. Then E = x~lEx < x~xCw{AIB{)x

=

Cw{AIB\x)

for each x e S. By Proposition 6.9 A < @xeS(A/Bix). From this embedding it follows that E < CG(A) = < 1 >, a contradiction. Thus Op(G) = < 1 >. 6.16. Corollary Let D be a Dedekind ZQ - domain, G an FC - group, A a just infinite DG - module which is D - torsion-free. IfCdA) = < 1 >, then G/£(G) is finite. Moreover, C,{G) includes a torsion-free subgroup of finite index. If charD = p > 0, then Op{G) = < 1 >. Proof By Theorem 6.15 G includes a normal abelian torsion-free subgroup of finite index. Since G is an FC - group, it is central-by-finite (see [TM, Lemma (7.5)]). The other assertions are either trivial or are contained in Theorem 6.15. 6.17. Corollary Let D be a Dedekind ZQ - domain, G a locally radical group, A a just infinite DG - module which is D - torsion-free. IfCdA) = < 1 >, then G includes a normal abelian torsion-free subgroup of finite index. Moreover, if charD = p > 0, then Op{G) = < 1 >. In fact, G is abelian-by-finite by Corollary 6.8. In particular, G is FC hypercentral. The next results shows that in some special situations it is possible to realize the reduction to the torsion-free case. 6.18. Proposition LetD be aDedekindZo - domain, G a group, x an element of infinite order of the center £(G), A ajust infinite DG - module, CG(A) =< 1 >. If A is D - periodic, then AnnD(A) = P e Spec{D). IfF = DIP, then A is F < x > torsion-free. Proof Corollary 6.4 yields that Anno(A) = P e Spec{D). Hence we can consider A as FG - module where F = DIP is a finite field. Put J = F < x > and think of A as a JG - module. Suppose that A contains an element 0 ± a s A such that / = Annj(a) * < 0 >. Since every non-zero ideal of J has finite index in J, aJ = JII is finite. It follows that x' e CQ{A) for some t e N, that is x' - 1 el. Since 0 * a e AnnA{I) < AnnA(x' - 1), this contradicts Corollary 6.5. This contradiction shows that A is J- torsion-free.

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Summing up these results, we deduce some consequences, which we simply state. 6.19.Corollary Let D be a Dedekind Zo - domain, G an FC -hypercentral group, the center of which contains elements of infinite order, A a just infinite DG - module, CG(A) = < 1 >. If A is a D -periodic module, then Anno(A) = P e Specify) and G includes a normal abelian torsion-free subgroup of finite index. Moreover, Op{G) = < 1 > where p = chariDIP). 6.20.Corollary Let D be a Dedekind Zo - domain, G a locally radical group, the center of which contains elements of infinite order, A a just infinite DG module, CQ(A) = < 1 >. If A is aD -periodic module, then Anno{A) = P e Spec(D) and G includes a normal abelian torsion-free subgroup of finite index. Moreover, Op(G) = < 1 > where p = char{DIP). The setting of these last results is that £(G) contains infinite cyclic subgroups and, in particular, f(G) * 1. We are finishing this chapter showing that this condition cannot be removed, as the next example shows. 6.21 Example Let p be a prime, C = Q C =< c„ \ {c„+\)p = c„, n e N >, < x > an infinite cyclic group, G = C X< x > wherex~ l c„x = c„+\,n e N. Let q be a prime such that q * p. Consider the group ring ¥qG. Suppose that / is a non-zero G - invariant ideal of the ring ¥qC. Since ¥qC = | J „ g N F ? < c„ >, there exists a number k such that lC\¥q < c* > = I\ * < 0 > . The factor - ring F ? < Ck > II\ is finite, so there is a number t e N such that c\ - 1 6 I\. In other words, a subgroup C contains an element c such that c - 1 e I\. If C\ =< c >G , then the index \C : C\\ is finite. Consider the ideal h of the ring ¥qC, generated by the elements b-\ where b e Ci. From the equation vy - 1 = (v - l)(y - 1) + (v - 1) + (y - 1) it follows that I2 < I. Since CIC\ is finite, ¥qC/I2 is finite and ¥qCII is finite too. Let R be a right ideal of ¥qG, generated by elements x" - 1, n e Z, and L be a right ideal of ¥qG with the properties: L > R and L * R.lfu e ¥qG, then u = a\C\xh + ... + a„c„x'", a, e ¥q, ci G C, tt e Z, 1 < i < n. It is clear that u = a\C\{xh

- 1) + ... + a„c„(x'n - 1) +a\C\ + ... + a„c„,

and therefore ¥qG = R + ¥qC, so that L = R + (L f] ¥qC). Since L=t R,h *< 0>. If 6 e 7 3 , then

Put h = L D F ? C .

Just Infinite Modules oxer FC-Hypercentral Groups x-mbxm

=

_(xm _ l)( JC -'")fc c '« +

75

fo^

that is i " * f e " e l . Since J e F,C, b = P\d\ +... + psds, where pj e F,, of, G C, 1 <j< s. Then jc^fec* = P\(x-mdixm) + ... + ps{x-mdsxm)

e F,C.

Hence x~mbxm e i f l F ? C = /3, which allow us to establish that h is a G invariant ideal of ¥qC. As we have already proved, FqC/h is finite, which gives that the index \VqG : L\ is finite too. Consequently, A = FqG/R is a just infinite F,G - module. It is easy to check that R f| F ? C = < 0 >, which assures that if 1 * c e C, then c - 1 £ /?. Suppose that c e CG(^I). Let g e G, then (g + R)c = gc + R = g + R, that is g(c - 1) e /?. Therefore g_1g(c - 1) e R, a contradiction. Thus CGOO n C = < 1 >. Since CG{A) is normal in G, it follows that Ca(A) = < 1 >. However, G is not abelian-by-finite.

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Chapter 7 Just Infinite Modules over Groups of Finite O-Rank

In the previous chapter we began to consider just infinite modules over an FC hypercentral group G. As we have proved, in this case G is abelian-by-finite. However, we obtained virtually no information about the structure of these modules. Now we will consider the case in which G is a FC- hypercentral group withfinite0-rank. In this case we may obtain some additional specific information about both the group G and the structure of just infinite modules over G. We shall need some assumptions about the underlying ring of coefficients. Actually, a Dedekind Z\-domain is a Dedekind Zo-domain in which Spec(D) is an infinite set. 7.1. Lemma Let D be a DedekindZi-domain, G a locally (polycyclic-by-finite) group of finite 0-rank, A a DG - module, which is D - torsion-free and A = MDG, for some finite subset M. Suppose that H is a finitely generated subgroup ofG of the same 0-rank, ro(H) = ro(G), and let B = MDH. If the factors AIAP are finite for P e 7r c Spec(D) and the set n is infinite, then ro{B) is finite. Proof Fix Pen: and put R = CH{AIAP). Since His polycyclic-by-finite, then R is finitely generated. Suppose that/? = < x\,... ,x, > andM= {a\,... ,a„}. There are elements uy € AP such that atXj = a, + Ujj, 1 < i < n, 1

<j
Letyi,}>2 e D be generators of P: P = y\D+yjD. Hence uy = Vyy\ +Wyy2, where v,j,Wij e A. Choose a finitely generated subgroup R\> H satisfying the condition v,y,wy e MDRi, whenever 1 < i < n,\ < j < t. Put E = MDR, E\ = MDR\ and

77

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Just Infinite Modules

£i = E\IE\P. Then we have a, + E\P = a, = (a,)*/, so that atDR = atD, 1 < i < n. This means that E = (E + E,P)IEyP = ZasM^R

= "Z^am

= £lsSia,Z>.

Since DIDP is finite, so is E. Since the index \R\ : R\ is finite, Lemma 1.14 implies that E\ is finite too. Now |/?i : H\ is finite, so that E\ is a finitely generated Z)//-submodule and [WJ 1, Theorem A] actually yields that E\ is a noetherian ZW-module. Let Y/BP be the P-component of E\IBP. By Lemma 1.9, AnnD(Y/BP) * < 0 > Clearly, £ < y. Take an integer / > 0 such that (JIBP)P' = < 0 > and call Y\IBP = (EilBP)P'. There is a D - submodule LIBP such that Ei/BP = (Y/BP) © (LIBP) [KI], and so (Y/BP) D (Y\IBP) = 0. We have already proved that E\IE\P is finite. Since E\ is Z)-torsion-free, Corollary 5.23 implies that Ei/EiPm is finite for any m e N (the finiteness of DIP implies that DlPm is finite, m e N). In particular, (E\IBP)I(Y\IBP) is finite, so that YIBP and BIBP are finite. Corollary 1.8 then gives that B e A(D,K\) for some finite subset K\ £ Spec(D). By Lemma 1.6 r D (B) = dimD/P(BIBP) for P i Ki. But n is infinite, so that n\it\ is infinite too. Since BIBP is finite for P e ;rWi, we may conclude that ro(B) is finite. Let D be a Dedekind domain. If A is a D-module of finite D-rank and M is a maximal Z)-free subset of yL then^/MD is a D-periodic. Put^o = MD and the set SPD(A) to be the set of all P e Spec(D), for which the P-component of AlAo is not bounded (that is its annihilator in D is zero). If B is another free Z)-submodule of A and ro(A) = rp(B), then AQ/(AO 0 B) and B/(Ao f) B) are finitely generated D-periodic modules, so AnnD(Aol(Ao P\B)) and AnnD(B/(A0 f]B) are non-zero, which shows that SPD(A) is independent of the choice of the free submodule Ao. In the sequel, we write Sp(A) instead of Spz(A). A module A is called a minimax module if it possesses a finite series of submodules with either artinian or noetherian factors. Let R be a noetherian ring, A a minimax i?-module. Take a finite series < 0 > = A0 to be a maximal ideal of D. By unique factorization of ideals, we have P" * P"+1, and then every cyclic torsion-free D-module cannot be artinian.

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79

Thus AIB has to be D - periodic and, since B is finitely generated, we may decompose B = T@C, where T is the periodic part of B and C is a finitely generated projective submodule of B (see, for example, [NW, Theorem 1.1,13]). Since AIB is periodic artinian, then UD(A/B) is finite, so UD(A/C) is finite too, because AnnD(T) * < 0 >. It follows that A e A(D,7t), where n = YlD(AIC) is a finite set of Spec(D). Hence, if A is a minimax £>-module, then SPD{A) is finite and, with the above notation, SPD(A) C HD(A/C). Furthermore, if A is torsion-free, we may choose a D-submodule C such that SPD(A) = TID(A/C). If D is a field, we simply note that a minimax D-module has finite dimension (as a £>-space). .4M abelian group A is called minimax, if the Z - module A is minimax. 7.2. Lemma Let D be a Dedekind domain, G a polycyclic-by-finite group and A a finitely generated DG-module of finite D-rank. If A is D-torsion-free, then A is a minimax D-module. Proof By Corollary 1.8, A e A(D,n), for some finite subset n of Spec(D). In other words, there exists a projective submodule C < A such that AIC is a n periodic module. In particular, ro(A) = ro(C). Take P e Y1D(A/C) and let YIC be the P-component of AIC, Y\IC = Q^^F/C). Since Y\ has finite Z) - rank, then dimDip(J\IY\P) is finite and, in particular, dimn/p(Yi/C) is finite. Then, YIC is an artinian £>-module by Lemma 5.6. Since IID(A/C) is finite, then AIC is artinian too. Thus A is D - minimax. 7.3. Corollary Let D be a Dedekind domain, G a polycyclic-by-finite group, A a finitely generated DG-module, M a finite subset A, H a subgroup ofG. IfroiA) is finite, then the submodule B = MDH is D - minimax andSpoifi) c: Spo{A). 1.4. Lemma Let D be a Dedekind domain, G a group, H,K normal subgroups ofG, A a finitely generated DG-module and Ma finite subset of A such that A = MDG. lfrD(MDH) andrD{MDK) are finite, then rD{MDHK) is finite. Proof Since ro{MDH) is finite then M includes a finite subset X such that (MDH)/(^laeM ^axD) is D - periodic. Similarly, K includes a finite subset Y a such that (MDK)/(£ eY yD) is £> - peniodic. The set FZis finite, therefore it is sufficient to show that (MDHK)/(Y> ^, azD) is D - periodic. Let a e M, h e H, g e K. There is an element u e D such that

{ah)u e Y,t

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and therefore (ahg)u G X) Let b G Mand consider the element b(xg~lx~x). Since AT is a normal subgroup of G, xg~xx~x e K. Thus there exists an element v e D such that bixgx-^v e E ^ j ^ a ^ Then

It follows that there is an element w e D such that {ahg)mv e X ^ ^ a z D .

D

7.5. Lemma Le? D be a Dedekind Z\ -domain, G an abelian group of finite 0-rank and A a just infinite DG - module. If A is D-torsion-free and charD = 0, then ro{A) is finite. Proof Let //be a finitely generated subgroup of G such that ro(H) = ro(G) and let M be a finite subset such that A = MDG. Put E = MDH. By Theorem 1.15, there is a subset n £ Spec(D) such that Spec(D)\n is finite and ^4 * AP for each Pen. Thus Lemma 7.1 and Lemma 7.2 yield that ro{E) is finite and £ is a minimax Z)-module; in particular, SPD(E) is finite. Since n is infinite, n\SpD(E) * 0. Consider J° e n:\SpD{E). Let/» = char(DIP) and Q/Hbe a Sylow p'-subgroup of G///. We claim that rD(MDQ) is finite. Since ,4A4P is finite, GICG{AIAP) is finite too. Put //i = CH(A/AP) so that /////i is finite. By Corollary 7.3, MDHX is a minimax /^-module and SPD(MDH\) C Spo(E). In particular, P g SpD(MDHi) and we may assume that / / = / / , , i. e.// < CG{AIAP). Let /<" be an intermediate finitely generated subgroup H < F < CQ{AIAP), and put B = MDF. Put M= {a\,... ,a„}, F = < x i , . . . ,xm >. Let jyi,y2 e D be generators of P: P = yiD+yzD. Then for some elements Ujj,vy G A, 1 < / < «, 1 <j < m we can write fif/Oy- 1) = M,y^i +Vij)>2 We choose a finitely generated overgroup L > F such that uy, v,y G C = MDL, 1 < i < n, 1 <j<m. Then

Just Infinite Modules over Groups of Finite O-Rank

81

at{xj - 1) = Uijy\ + Vjjy2 e CP,

so FL-module and, by Lemmal.9, AnnD(Ci/DP) * < 0 >. Suppose that P' = AnnD{CJBP). Then C/fiF = Ci/5/> © C2/BP ([KI]), so CP' < Ci, and B f] CP' < B n C2 = BP, because 5 < Ci. By Lemma 6.14, F < CG(CP7CP'+1), for each / € N. It follows that FP'~l < CG{CICP') and thus FP'~l < CG{BIBP). Let S be a transversal to H in F. Then S is finite and we may be expressed as B - ^xsS Ex. Put B = BIBP and E = {E + BP)IBP. Since F/// is a /?'-group, for every x e S there exists rx e N such that (rx,p) = 1 andxr* e //; in particular Fjcr* = E, that is £xr* = E. We have already proved that Exp'~ = £, and since rx and /? are coprime, we deduce that Ex = E. Then B = y , „ £y = £. From the £> - isomorphism E = (E + BP)/BP = E/(Ef]BP) and the inclusion EP < EHBP it follows that dimDip(E) < dimDip{EIEP). Since B = J ^ ^ i u c , P £ Spo(E) implies that P e SpD{B). Thus P 0 SpD{E). Lemma 1.6 implies that r^GB) < rD(E) which forces the equality: r^{B) = r/j(£). Let £ = {FxlH [ A e A} be a local system of finitely generated subgroups of Qi/H= (QD CG(AIAP))IH. Thus MDQX

=^X^M(DFX).

The arguments above show that rD(M(DF{) = ro(E) for each X e A, we conclude that ro(MDQ\) = ro(E). Since |(2 : £hl is finite, reasoning in the same way, we find that m{MDQ) = ro{E) is finite, as claimed. Since n is infinite, p) P e AP has infinite index in A, which gives ("Ipe*^ = < 0 >. As a consequence, we deduce that the characteristic of the finite fields DIP, Pen, cannot be a constant. Otherwise, if char(D/P) = p for each P e n, A could be embedded in the Cartesian product Ylp^AlAP, so pA = 0, while charD = 0. Therefore, there exist Pi,P2 £ ff such that char(D/P\) * char{DIPi). Further, we may assume that Pi,P2 £ Spn(E). For / = 1,2, let Qj/Hbe a Sylowp,' - subgroup of GIH. Then G = 0 i 2 2 and, since we have already proved that roiMDQi) are finite, then A"D(^) is finite by Lemma 7.4, as required. The above lemmas are the reformulated versions of some results of [ZKT]. 7.6. Theorem Let D be a Dedekind Zi-domain of characteristic 0, G an FC-hypercentral group offinite0-rankandA a just infinite DG-module which is D - torsion-free, CG(A) = < 1 >. Then ro{A) is finite and G includes a torsion-free abelian normal subgroup of finite index. Moreover, ifK is the field of

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fractions ofD andn = ro(A), then G is isomorphic to an irreducible subgroup of GL„(K). Proof By Theorem 6.15, G includes a torsion-free abelian normal subgroup H of finite index and, by Proposition 6.9, there exists a ZW-submodule B such that AIB is a just infinite DH - module. If X is a transversal to H in G, it follows that AlBx is just infinite for every x e X, and A embeds in @xeXA/Bx. By Lemma 7.5, each summand of the above direct sum has finite Z)-rank. Since X is finite, ro(A) = n is finite. If E = A ®o K, we note that n = dim^E and CG(E) = CG(A) = < 1 >, so that G can be considered as a subgroup of GL„(K). This subgroup is irreducible, because E is an irreducible ATG-module. 7.7. Corollary Let G an FC-hypercentral group of finite 0-rank and A a torsion-free just infinite ZG-module with CG(A) = < 1 >. Then (1) G is a finitely generated abelian-by-finite group; (2) the additive group of A is a minimax. Proof By Theorem 7.6 G is an abelian-by-finite irreducible linear group over Q and [CV 3] implies that G must include a finitely generated free abelian subgroup of finite index, then showing (1). (2) follows from Lemma 7.2. 7.8. Corollary [KK] Let G a locally radical group of finite 0-rank and A a torsion-free just infinite ZG-module with CG(A) = < 1 >. Then (\)Gis a finitely generated abelian-by-finite group; (2) the additive group of A is a minimax. Proof

It suffices to apply Corollary 6.8 and the above result.

Let G be an abelian-by-finite group of finite 0-rank and let H be an Abelian subgroup of G of finite index. In the next results, we shall put Sp(G) = Sp(H). 7.9. Theorem Let D be a Dedekind Z\-domain with charD = p > 0, G an abelian-by-finite group of finite 0-rank, A a just infinite DG-module, CG(A) = < 1 >. If A is D - torsion-free andp £ Sp(G), then ro(A) is finite. Moreover, G is isomorphic to an irreducible subgroup of GL„(K) where K is the field of fractions for D, n = ro(A). Proof As in the proof of Theorem 7.6, it suffices to assume that G is abelian. Let / / b e a finitely generated subgroup of G such that ro(G) = ra(H) and let M b e a finite set of generators of A : A = MDG. Since p <£ Sp(G), the Sylow p-subgroup of GIH is finite. In other words, there exists an overgroup Q> H such that QIH is ap'-group and \G : H\ is finite. A verbatim repetition of the arguments given in Lemma 7.5 and in Theorem 7.6 proves the finiteness of ro(A) and the

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required embedding of G. Given a group G; if A is a Z)-periodic just infinite DG-module, then, by Corollary 6.4 Anno(A) = P e Spec(D). If there exists an element x e £(G) having infinite order, then by Proposition 6.18 A is F < x >-torsion-free, where F = DIP is a finite field. Thus, we come back to the torsion-free case where D = F < x > is the group-ring of an infinite cyclic group < x > over a finite field F. In this case, the following lemma is almost obvious. 7.10. Lemma Let D = F < x >, where F is a finite field, \F] = q, < x > is infinite. Given k e N,we consider the map ). 7.11. Lemma [KTZ] Let F be a finite field of characteristic p, G an abelian minimax group with Sp(G) = {p} and A a cyclic FG-module such that AnnFG(A) is a prime ideal of the ring FG. Given 1 * z e G and an infinite cyclic group < t >, ifD = F < t >, then A can be viewed as a DG-module defining at = azfor each a e A, and, with this meaning, there exists a finite subset n £ Spec(D) such that A e A(D,n). Proof If / = AnnfG(A), then A = FGII. Since / is a prime ideal, the factor-ring R = FGII is an integral domain. Then the mapping

...,

whose union L = |J„eN-Rn n a s a n automorphism q>*, given by * induced an isomorphism between £2/^1 and E1/E0. Corollaryl.8, taking in account that H is finitely generad, implies the existence of a finite subset n £ Spec(D) such that EO,EIIEQ e A(D,n). Hence t\ = t,D\ = F < t\ >,(n\)q>* = n, and we get £2 IE\ e A(D\,n\). In this case,

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Lemma 7.10 yields that EilE\ e A{D,n). Using similar arguments and simple induction we can prove that E„+\IE„ e A(D,n) for every n 6 N. Lemma 1.2 proves that and L e A(D, n), and hence R e A(D, n) by the same Lemma 1.2. 7.12. Corollary [KTZ] Let F be a finite field of characteristic p, G an abelian minimax group with Sp(G) = {p}, and A a noetherian FG-module. Given 1 * z e G and an infinite cyclic group < t >; if D = F < t >, then A can be viewed as a DG-module defining at = azfor each a e A, and, with this meaning, there exists a finite subset n c Spec(D) such that A e A(D,n). Proof

Let M be a finite subset such that A = MFG. We have the embedding FG/AnnFG(A)

<

Y\a<MFG/AnnFG(a)

and (a)FG = FGIAnnFaia). Since (a)FG is noetherian, FG/Annpcia) is a noetherian FG-module too. Thus R = FGIAnnFaiA) ' s a noetherian ring. Considering A as an 7?-module, A has a finite series of submodules < 0 > = AQ < A\ < ... < A„ = A and every factor A ,A4,-i = R/Eh £ , is prime (see [SR, Theorem 9.40]), 1 < / < n. If £/, is the canonical preimage of E, in FG, then {/, is a prime ideal and AJAt-\ = FG/Uj. By Lemma 7.11, there exists a finite subset Kj c Spec(D) such thaty4,A4,_i e ,A(,D,7r,), 1 < i < n. By Lemma 1.2, it suffices to define ^ = Ui = Ao < A\ < ... Aa < Aa+\ < —Ay = A such that ^ 0 + I ( J C - 1 ) < Aa for every x e G,a< y.In other words, Aa+\{coRG) }. It is easy to see that C,RG(A) is an RG - submodule of A.

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85

We can construct the upper RG - central series ofA : < 0 > = Co < C\ < ... Ca < Ca+i < ... Cy where Ci = t,RG{A),Ca+\ICa = ^Ra{AICa),a < y,^RG(A/Cr) =< 0 >. The last term of this series is called the upper RG - hypercenter of A. A module A is RG - hypercentral if and only ifA = C y . Let ,4 be an RG - nilpotent module, < 0 > = Ao , AH"~X * < 0 >. Put B0 = < 0 >,BX = AH"-l,...,B^i

= AH,B„ = A.

Clearly the series Bo < Bi < ... < £„ = A is RG- central. 7.13.Theorem Let F be a finitely generatedfield,A a finite dimensional vector space over F.IfG is a soluble automorphisms group ofA, then G has a series of normal subgroups < 1 > < H < E < G, where (1) G/E is finite; (2) EIH is a countable free abelian group; (3) H is a nilpotent subgroup; (4) if char F = 0, then H is torsion - free; if charF = p > 0, then H is a boundedp - subgroup; (5) A is an FH - nilpotent module. Proof We can consider G as a subgroup of GL„(F) where n = dimpA. By Maltsev's theorem (see, for example, [WB, Theorem 3.6]) there are an element g e GL„(F) and a normal subgroup £o of G such that the index \G : Go\ is finite and g'l(Eo)g< T„(F) where F is an algebraic closure of the field F. Let g = ||ay||i<,j<„ and F\ = Ft0?;!! - 'J - "]• Then F\ is a finite field extension of F and g~lEog < T„(F\). In particular, the field F\ is finitely generated too. Put A\ = A%F F\, E\ = g~lEog, Hi = Ei f] UT„{Fi). Then Ai has a basis in which the matrix of every automorphism h e Hi is unitriangular, moreover, this basis is the same for all elements h e Hi. In other words, Ai has a series of FHi -

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submodules < 0 > = Co < Ci < ... < C„ =Ai such that Ctih - 1) < C,-_i for every he Hi, I < i < n. Put H = gHig~l, Z, = Cjg~x, 0 < / < n. Then H < EQ and for every element y = ghg~l £ / / w e have Zt(y-l)

= Clg-i(ghgrl

- 1) = Cig-lg(h-l)g-1

= C,{h-\)g-i

< C^g'1

= Z,_,

This means that the series < 0 > = Z 0 < Z i < ... < Z„ =Ai is F i / / - central. In other words, Ai is F i / / - niipotent, so that A is an FH niipotent module. The group UT„{Fi) has a central series UT„(Fi) = UT$\Fi)

> UTf\Fi)

> ... > UTin)(Fi)=<

1 >,

where UTim)(Fi)

= < t,j(a) | a e Fuj - i > m >, t0{a) =

E+aE0,

(see, for example,[KMM, 16.1.2]). Moreover, U&\Fi)IUlt+l){Fi)

=F\ x ... x F\

(here each factor F\ is the additive group of the field Fi, [KMM, Ch. 4]). In particular, Hi is niipotent. Moreover, if charF = 0, then F\ is torsion - free, so that Hi is torsion - free. If char F = p > 0 then F\ is an elementary abelian p group, so that Hi is a bounded p - group. Since H and Hi are conjugate, the same is valid for subgroup H. Finally, T„(Fi)IUT„{Fi)

= U(Fi) x ... x U(Fi) .

Theorem 4.10.1 from [KG] yields that U(Fi) is a direct product of finite cyclic subgroup and countable free abelian subgroup. It follows that Eo includes a G invariant subgroup E such that \G : E\ is finite and EIH is a countable free abelian group.

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7.14. Corollary Let F be a finitely generated field, G a soluble group, A a simple FG - module, Ca(A) = < 1 >. IfdimpA is finite then G is a finite extension of a countable free abelian subgroup. Proof

By Theorem 7.13 G has a series of normal subgroups <1>
where GIE is finite, EIH is countable free abelian, H is nilpotent and A is an FH nilpotent module. Put A\ = ^FH{A). Since H is normal in G, A\ is an FG submodule. Since A\ * < § >, A = A\.\a. other words, for any a e A, h e Hvte have a(h - 1) = 0, i.e. h is an identity automorphism. Hence H = < 1 >. 7.15. Corollary Let R be a finitely generated integral domain, A an R - module of finite R - rank which is R - torsion - free. If G is a soluble automorphisms group of an R - module A, then G has a series of normal subgroups <\> 0, then H is a boundedp - subgroup; (5) A is an RH - nilpotent module. Proof Let F be the field of fractions of the ring R. Then F is a finitely generated field. Put B = A<S>RF. We can uniquely extend the action of G on A to an action of G on B. Since rR(A) is finite, dimpB - r^iA) is finite too. Now we can apply Theorem 7.13. 7.16. Corollary [ZKT] Let F be afinitefield,< t > an infinite cyclic group, J = F < t >, A a J - module of finite J - rank which is J - torsion -free. If G is a soluble automorphisms group of the J - module A, then G has a series of normal subgroups <\>
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(4) A is a JH - nilpotent module. l.YI. Corollary [CV 3] Let F be an algebraic number field, A an finite dimensional vector space over F.IfG is a soluble automorphisms group of A, then G has a series of normal subgroups <\> an infinite cyclic group, D = F < x >,G an FC-hypercentral group of finite 0-rank and A a just infinite DG-module which is D - torsion-free, CG{A) = < 1 >. Then (1) G is a finitely generated abelian-by-finite group. (2) A is D-minimax. Proof By Theorem 6.15 and Proposition 6.9, we may assume that G is a torsion-free abelian group of finite 0-rank. Choose a finite set M of generators for A: A = MDG. Let H be a finitely generated subgroup of G such that ro(G) = ro(H) and let QIH be the Sylowp'-subgroup of GIH, where p = charF. As in Lemma 7.5, we can prove that B = MDQ has finite D-rank, which gives that Q is finitely generated by Corollary 7.16. In other words Sp(G) = {p} and, replacing G by G x< x >, by Corollary 7.12, we have A e A(D,n), where n is a finite subset of Spec(D). Then, it suffices to apply Lemma 7.2. 7.19. Corollary [KK] Let F be a finite field, < x > an infinite cyclic group, D = F < x >, G a locally radical group of finite 0-rank and A a D-torsion-free just infinite DG-module with CG(A) - < 1 >. Then ( 1 ) G I J a finitely generated abelian-by-finite group. (2) A is D-minimax. 7.20. Corollary Let D be a Dedekind Zo-domain, G an FC-hypercentral group whose center contains an element x of infinite order and A a D-periodic just infinite DG-module with CQ(A) = < 1 >. Then (\)AnnD(A) = P e Spec(D). (2) G is a finitely generated abelian-by-finite group.

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(3) A is a torsion-free F < x >-minimax module, where F = DIP. As the results of this chapter indicate, the most interesting case is the abelian one, in which we may perform the following construction (see [RZ]). 7.21. Construction [RZ] Let R be an integral noetherian domain, in which every non-zero prime ideal has finite index (for example, a Dedekind domain). Consider an abelian group G and a just infinite /?G-module A. For each x e RG define the mapping ix : A —• A by the rule: a(ix) = ax,a e A. Clearly, ix is an RG - endomorphism of A, and the mapping O: x —• ix, x e RG, is a homomorphism of the ring RG in the ring EndRG(A). Furthermore, Kerd> = AnnRG(A). Put K = RGII where / = AnnRG{A). By Corollary 6.2 EndRG(A) has no zero-divisors, i.e. / is a prime ideal of RG, and hence K is an integral domain. Let 0 * a e A. Consider the mapping 8: K —• A, defined by the following rule. Let x e K, then x = x +1 for some x e RG. Put now x6 = ax. If x = x\ +1, then x\ = x + z where z e AnnRo(A), so that this definition is correct. It is easy to prove that 6 is an RG - homomorphism. Let x e Kerd, then ax = aix = 0, that is a e Kenx. However, by Lemma 6.1 Kenx = < 0 >. This means that KerO = < 0 >, that is K = Imd. Since ImO is an RG- submodule of A, ImG has finite index in A. If L is a non-zero ideal of K, then L6 is a non-zero RG - submodule of A. Thus AILQ is finite, in particular, ImOILd is finite, so that KII is finite too. Consequently, a group ring RG includes a prime ideal / such that K = RGII is a just infinite /?G - module and A is a finite essential extension of K. Question 4 Let F be afield, G a metabelian group of finite section rank {even a minimax group). Describe the structure of just infinite FG - modules.

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Chapter 8 Just Infinite Modules over Polycyclic-by-Finite Groups

Chronologically polycyclic-by-finite groups have formed the first class of groups, over which just infinite modules have been studied. D. J. S.Robinson and J. S. Wilson in their fundamental paper [RW] showed that the classification of just non-polycyclic groups heavily depends of the behavior of just infinite modules over polycyclic groups (even the term "just infinite module" has arisen in this work ). All subsequent papers devoted to various types of just non-^-groups showed that the study of just infinite modules is a necessary important component of this research. Let G be a polycyclic group, D a Dedekind Zo - domain and A a just infinite DG - module. If A is D - torsion - free, then by Corollary 6.8 GICG{A) is an abelian-by-finite group, and in this case A is D - minimax. If A is D - periodic, then by Corollary 6.4 Anno(A) = P e Spec{D) and we can consider A as an FG module where F = DIP is a finite field. This case was considered first by D. J. S.Robinson and J. S. Wilson. We are going to present their main results in this direction. Note that Proposition 6.9 and similar results allow us to realize the reduction to normal subgroups of finite index (see [RW]). A polycyclic group G is said to be primitive if G satisfies the following conditions'. (1) G is torsion-free; ( 2 ) « G ) = FC(G); (3) ifH is a subgroup such that the set {Hg \ g e G} is finite, then H/CoreG(H) is finite; (4) ifG is metabelian-by-finite, then G is metabelian; (5) ifG is abelian-by-finite, then G is abelian;

91

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(6) G = (Fitt(G))L where L is a nilpotent subgroup. 8.1. Lemma [RW] Every polycyclic-by-finite primitive subgroup of finite index.

group includes a characteristic

Proof Note that G includes a normal torsion-free subgroup G\ of finite index (see, for example, [SD 2, l.C]). Let s = \G : Gi\, then G2 = Gs < G\. Thus G2 is a characteristic torsion-free subgroup of finite index. The subgroup FC{G) is finitely generated, hence the index \G : CQ(FC(G))\ is finite. Put G 3 = CG{FC{G)). Theorem C2 from [RJ 2] yields that G includes a characteristic subgroup Gt, with the property (3). If G is abelian-by-finite then we choose an abelian characteristic subgroup Gs of finite index; if G is not abelian-by-finite, then put Gs = G. If G is metabelian-by-finite, choose in G a characteristic subgroup Gs of finite index; if G is not metabelian-by-finite, then put Gs = G. Now put G6 = Gj n / / 3 C\ G4 fl Gs • The subgroup G6 includes a characteristic subgroup of finite index Gi such that Gj = (Fitt(G(,))L\ where L\ is a nilpotent subgroup (see, for example, [SD 2, 3.6, Theorem 3]). Since \G : Gj\ is finite, G7 includes a characteristic subgroup Go > Fitt(G(,) such that \Gj : Go\ is finite. Therefore Go = (Fitt(Ge))L where L = Go fl L\ is a nilpotent subgroup. From now on, we are considering just infinite FG - modules, where G is a primitive polycyclic group and F is a finite field. In this case it is worth mentioning that the condition (JI 2) from the definition of the just infinite module is automatically satisfied: indeed by [PD 1, Theorem 12.3.7] a simple FG module is finite. Let G be a polycyclic-by-finite group. A subgroup P is called a plinth of G if the following conditions are sufficed: (PL X)P is a non-identity torsion-free abelian subgroup ofG; (PL 2) if H= NG(P), then the index \G : H\ is finite and HICH(P) is abelian-by-finite; (PL 3) if S is a subgroup of H having finite index in H, then P ® zQ is a simple QS - module. (See, for example, [PD 1, Chapter 12, Section 3]). Every polycyclic-by-finite group includes a plinth (see, for example, [PD 1, Lemma 12.1.4]). Let G be a primitive polycyclic grou. Choose a non-identity free abelian normal subgroup P of G of the smallest possible rank. Replacing P by a larger subgroup if necessary, we may assume that P is contained in none abelian normal subgroup of G as a proper subgroup of finite index. Let R < P and suppose that \G : NQ(P)\ is finite. Since G is primitive, PJCorea(R) is finite, and therefore \P : Corea(R)\ is finite too. Since P<8> z Q is a simple QG - module then

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93

G/CG(P) is abelian-by-finite (see, for example, [WB, Lemma 3.5]). This means that P is a plinth of G. Consequently, a plinth of a primitive polycyclic group is a normal subgroup. Since the case of the group G with f (G) * < 1 > has been previously considered, we can consider only the case of the group G with £(G) = < 1 >. 8.2. Lemma [RW] Let F be a finite field, G a primitive polycyclic group with £(G) = < 1 >, P a plinth ofG, P is normal in G,R = FP. If A is a just infinite FG - module with Ca(A) = < 1 >, then A as an R - module is torsion-free and has finite R - rank. Proof Suppose that A has an R - periodic element a * 0. Then Annn(a) * < 0 >. We have a e AnnA(Annn(a)). Let •A4 = {Q | Q is a non-zero ideal of if such that Ann A{Q) * < 0 >}. Then M =£ 0. The group ring R is noetherian (see, for example,[PD 1, Theorem 10.2.7]). It follows that M has a maximal element /. Put B = AnnAif). Let I\,h be ideals of R such that I < h, I < h and l\h = I- From B(I\h) =< 0 > we obtain that either AnriA(I\) * < 0 > or AnnA(Ji) * < 0 >. By the selection of/, either is / = h or / = h- In other words, / is a prime ideal of/?. If x e G, then Bx = AnnA(Ix). In fact, let B\ = AnnA(Ix). If b e £,>> e 7, then

(&x)(x~1.y*) 1

=

(fyO* = o>

x x

that is fix < Tii. Hence (fii)x" < AnnA((I ) ~') = B, so fix = fii. If x e NG(I), then fix = AnnA(Ix) = AnnA{I) = B, thus we can consider B as an F(NG(I)) - submodule. Let T be a transversal to MsCO in G. Then B(FG) = ^ gT-^x- Suppose that this sum is not a direct sum. Then there are elementsx\,... ,X£, Xk+\ £ Tsuchthat Bxi+i n (fixi + ... + Bxk) * < 0 >. Put><, = x,(x^j), 1 < i < k. Clearly, y, £ NciT) for any z. Then fin(fiyi

+ ... + B y * ) * < 0 > .

Since ArmA{I+ (/>" n ... n / w ) = ^Iw^CO 0 (AnnAUy')

+ ... + AnnA(P"))

=Bf](By

from the choose of / we obtain the inclusion H I - S K * ^ ' - ^' anc * t n e r e f ° r e Iy'...In < I. Since / is a prime ideal, P" < I for some i. However, in this case

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Just Infinite Modules

I < P~> . Taking in account that Ann A{F ) = ByJ1 =t=< 0 >, by the choice o f / w e can conclude, that / = pi . This implies that yt e NG{I), a contradiction. Consequently, B(FG) = ®xsTBx. Put H = NG(I). Assume that B includes a proper non-zero FH - submodule C. Then C(FG) = (BxeTCx. Since A is a just infinite FG - module, AIC(FG) is finite. This means that the set T is finite. In other words, the index \G : H\ is finite. The subgroup P is a plinth in H too. By Bergman's Theorem (see, for example, [PD 1, Corollary 9.3.9]) the factor-ring R/I is finite. It follows that there exists a number m eN such that 1 -ym e I for each y e P, that is Pm < CG(B). Since Pm is normal in G, pm

= x-\pmx

< X-1CG(B)X = CG(Bx)

for every x e T. It follows that /"" < CG{B{FG)). Since the A/B(FG) is finite, there is a number mi e N such that /""' < CG(A/B(FG)). Since the additive group of ^ is an elementary abelian p - group, there is a number mj e N such that /"" 2 < C G 0 4 ) =< 1 >, and we obtain a contradiction (G is torsion-free). If we assume that B is a simple FH- module, then B is finite (see, for example, [PD 1, Theorem 12.3.7]). Thus there is a number m 3 e N such that Pm e CG{B), and again we come to a contradiction. Consequently, A is R - torsion-free. By Theorem C from [RJ 1] there are a free R - submodule E of A and a non-zero ideal A of R such that every element oiAlE is annihilated by some product A*' ... A*" for suitable x\,... ,x„ e G. Since P is a plinth of G, there is a maximal ideal L of R which includes no conjugate of A by Theorem E from [RJ 1]. Corollary CI from [RJ 1] yields that^ = E + AL and EL = EC\AL. In particular, AIAL and EIEL are isomorphic as R - modules. Since L is maximal in R, the factor-ring RIL is finite, so P/Cp(PJL) is finite. Hence for some / G N the ideal L generated by all elements xl - l,x e P, lies in L. It is clear that L is a non-zero G - invariant ideal of R. Therefore AL is a non-zero FG - submodule of A (A is R - torsion-free). It follows that AIAL is finite, hence AIAL is finite too. EIEL is finite sinceAIAL =R EIEL. The R - module E is free, that is E = ® a s r £« where Ea = R for every a e T. Then jE/isZ, = ® a e r EJEaL where EJEaL = RIL. It follows that T is finite. In other words, the R - module E has finite R - rank. Since P is normal in G, A*1... A*" < R. that is A*1... A*" < AnnR(A/E). Thus ,4/JE is /? - periodic, and therefore the R - module A has finite R - rank. 8.3. Lemma [RW] Let F be a finite field, G a primitive polycyclic group with C,(G) =< 1 >, P a plinth ofG and also P is normal in G. Let A be a just infinite FG - module with CG(A) = < 1 >. Then P is a maximal normal abelian subgroup ofG. Proof

Put R = FP. Suppose that there is a normal abelian subgroup P\> P

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95

suchthatPi * Pandlet^i = FP\. Let A = AnnRl(A). By Lemma 8.2 A * Ru It is clear that A is G - invariant. Let x e R\\R. Then x induces an R - linear mapping in A. If a e A, then aR<x>=R<x>

/AnriR<x>(a).

Since the R - module^ has finite R - rank, AnriR<x>{a) * < 0 >. If a\,... ,ar are the maximal R - independent subset of A, then P| 1{ai) = AnnR^iA) since A is R - torsion-free by Lemma 8.2. This means that A * < 0 >. Suppose that there are G - invariant ideals I\ and h of R\ such that I\h < A, but I\ $ A, h $ A. If y e I2, then ^(/i>0 = < 0 > . Thus the mapping a + AI\ —• ay,a e A, is an R\ - homomorphism of AIA(I\) with the image Ay. Since I\ S A and I\ is G - invariant, ^(/i) is a non-zero FG - submodule of A. It follows that AIA(I\) is finite, so Ay is finite too. The ring R\ is noetherian, so there are elements y\,... ,y, e h such that h = Riy\ +...+R\yt. Therefore A(h) = Ay\ +...+Ayt is a non-zero finite FG - submodule of A. This contradiction shows that such ideals I\ and h do not exist. By Lemma 5 of [RJ 2] A = Ai fl ... fl Am where Ai,... , Am is a complete G - orbit of the prime ideal of Ru Let K = (1 + Aj) n Pi. Then { F | g e G } = {(1 + A,) n ^1 | 1 < i < m}. Since G is a primitive polycyclic group and the index \G : NQ{K)\ is finite, KICoreG(K) is finite. If u e CoredK), then M - 1 e Ai D - fl Am = A, so /l(w - 1) = < 0 >. Since CG(A) = < 1 >, Corea(K) = < 1 > . Hence K = < 1 >, since G is torsion-free. Consequently, Ai is a prime ideal of Ri such that the index \G : NG(AI)\ is finite and (1 + Ai) n Pi = < 1 >. By Theorem D from [RJ 2] Ai = (Ffl Ai)Ri, so that Ai = < 0 >. In particular, A = < 0 >, a contradiction. 8.4. Lemma [RW] Let F be afinitefield, G a primitive polycyclic group with C(G) =< 1 >, P a plinth ofG and also P is normal in G. Let A be a just infinite FG - module with CG(A) = < 1 >. Then G splits over P : G = P X T where T is a free abelian subgroup, CT(P) = < 1 > and P ®z Q is a simple QT- module. Proof Put R = FP and let K be a field of fractions of R, B = A®RK. By Lemma 8.2 dimKB is finite. Put H = CG(P)- Then P possesses a finite series < 0 >= Bo < Bi < ...

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HIHX < H/CH(Bi/B0)

x ... x

H/CniBr/Br-x)

we obtain that HIH\ is abelian-by-finite. There is a number k e N such that [Hk,Hk] = H2 < H\. Since the additive group of B is an elementary abelian p group, p = charF, there is a number i i e N such that 77*' - CG(B) = < 1 >. It follows that Hj = < 1 > since G is torsion-free. Thus Hk is abelian. Since Hk < CG(P), HkP is an abelian normal subgroup of G. By Lemma 18.3 Hk < P, i.e. H is central-by-finite. By Schur's Theorem (see, for example, [RD 19, 10.1.4]) [H,H] is finite, and hence [H,H] = < 1 >. Lemma 8.3 yields that CG{P) = H=P. Since P < Fitt(G), C = £(Fitt(G)) < CG(P), thus C < P. It follows that PIC is finite, since P is a plinth of G. Since Fitt(G) is torsion-free, Fitt(G)IC is torsion-free too (see, for example, [RD 19, 5.2.19]). This means that P = C, therefore P = F/«(G) because P = CG(P). Let T be a nilpotent subgroup with the property G = PT. Suppose that P f)T * < 1 >. Since POT is normal in 7, L = f (7) n P * < 1 >. Since G = P77, the subgroup L is normal in G. Then P/Z, is finite. It follows that [P, 7] is finite, thus [P, T\ =< 1 > and 7" < C G (P) = P, a contradiction. It shows that P f] T = < 1 >. Since P is a normal abelian subgroup of the smallest possible rank, P ®z Q is a simple Q 7 - module. In this case T/CT(P) is abelian-by-finite (see, for example,[WB, Lemma 3.5]). Hence T is abelian-by-finite. It follows that T is abelian, because 7 is torsion-free and nilpotent. From Lemmas 8.2 - 8.4 we obtain 8.5. Theorem [RW] Let F be a finite field, G a primitive polycyclic group with C,(G) =< 1 >, A a just infinite FG - module with CG(A) = < 1 >. Then (1) P = Fitt(G) is an abelian subgroup; (2) P is a plinth of G; (3) G = P X T, where T is abelian, CG(P) = P, P ®z Q is a simple QT module; (4) the FP - module A is torsion-free and has finite R - rank. The aforesaid paper also describes a method for constructing just infinite modules, which we recall now. Let G = P X T be a primitive polycyclic group, P = Fitt(G) a plinth of G, CG(P) = P, P and T finitely generated torsion-free abelian subgroups, P ®% Q a simple Q 7 - module. Put R = FP and let .K be the field of fractions of R. If r e N, then let AT(r) be a /^ - vector space formed by all r - tuples over K and let R^ the corresponding R - submodule of all r - tuples over R. The action of elements of T on P by conjugation may be extended in the obvious way to the actions on R and K. Then Tacts on P w and K^ through its action on components, and these action can be extended to GLr(K) through its entries.

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97

Now choose a derivation from Tto GLr{K), that is a function 8 : T — GLr(K) with the following condition: (txh)s

= (r?)"(f2)*

forallfi,<2 e 71. Define a new action of T on X'W by the rule vt = v't5,v e £to, f e T. In a natural way, P acts on £ w by multiplication. It is an easy matter to check that this gives in K^ the structure of an FG - module. Define M(S) to be the FG - submodule of K^ generated by / ? w , that is M{8) = ^ileTR^t. is a finitely generated FG - module. Furthermore, M(S) as an R - module is torsion-free and has finite R - rank. 8.6. Theorem [RW] Let F be a finite field, G = P X The a group satisfying (1) - (3) of Theorem 8.5, A an FG - module which is R - torsion-free and has finite R rank. Then A is a just infinite FG - module if and only if A is a finite extension of a just infinite module isomorphic to M{8) for some derivation 8: T —• GLr(K) where K is the field of fractions of the ring R. Proof One implication here is trivial: indeed an R - torsion-free finite extension of any just infinite FG - module B is always just infinite. It happens because a non-zero submodule L of A must satisfy L f| B * < 0 >; thus both \B : B n L\ and \A : L\ are finite. Conversely, assume that A is just infinite. We regard A as embedded in V - A ®A K which is an r - dimensional K - vector space. The action of T on A extends naturally to Fif we define (a ®f)t = (at) ®f where a eA,fe

K,t e T.

It is easy to check that according to this definition V becomes an FG - module. Choose a basis {e\,...,er} of V consisting from elements of A. For each t e T we have

where fyit) e K. Put

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ts = MOWThen ts e GLr(K), because the elements e\t,... ,ert are linearly independent. Let v e V,v = Yji< w n e r e hi e K. Then

Write [v] for the coordinate vector of v. Then [vt] = [v]'ts. Hence [v]'ii(tit2)s

= [Wife] = [v
for all v e K, (i,( 2 £ T. It follows that C1/2)* = (A)<ns2 so that 5 is a derivation from Tio GLr(K). We make ofK^ if put

into an FG - module

at = a't5, a e A, t £ T. Then from [vt] = [v]'(td) we obtain that the mapping 6 : v -+ [v] is an FG module - isomorphism from Fto K^. Since e, e A,M{8) < Ad. Of course M(5) is just infinite and has finite index in AG The above result raises the question on the necessary and sufficient conditions for any M{8) to be just infinite. In this setting, the following result is very useful. 8.7. Lemma [RW] Let F be a finite field, G = P X T be a group satisfying (1) (3) of Theorem 8.5, A an finitely generated FG - module which is FP torsion-free. Then A is a just infinite FG - module if and only if the following conditions hold: (1) ifK is the field of fractions of the ring R = FP, then A<&RKis a simple KT - module; (2) for each m eNthe factor-module A p- = AIA{a>F(Pm)) is finite. Moreover, if for some n e N every finitely generated R- submodule lies inangeneratedR - submodule ofA, then (2) is automatically satisfied. Proof

Put V = A <S)R K where R = FP and regard A as an FG - submodule of V.

Just Infinite Modules over Polycyclic-by-Finite Groups

99 m

Suppose first that A is just infinite. Since A is R - torsion-free then A(a>F(P )) is a non-zero FG-submodule for each m e N. It follows that A/A(coF(Pm)) is finite. If Vo is a non-zero %T- submodule of V, then A fl Vo is a non-zero FG - submodule of^4. Hence A/(A f\ Vo) is finite and therefore Vo = K. Conversely, assume that conditions (1) and (2) are satisfied. Since A is finitely generated over FG and the ring FG is noetherian (see, for example, [PD 1, Theorem 10.2.7]), A is a noetherian FG - module. Let M = {U | [/ is an FG - submodule such that AIU is infinite }. It is clear that M * 0. Then ,M has a maximal element B. Let 5 = {U | [/ is an FG - submodule such that U > B and U * B}, B0 = f\ S. Then AIU is finite for every U e S. If B = Bo, then A/B is a just infinite module. If Bo * B, then A/Bo is finite. Hence Bo/5 is an infinite simple FG - module. However, for every polycyclic group G and every finite field F each simple FGmodule is finite [RJ 1, Theorem A]. This contradiction shows that A/B is a just infinite FG - module. Suppose that B * < 0 >. If A/B is not R -periodic, then B\IB be the R - periodic part of A/B. Clearly, B\ is an FG - submodule of A. Since B\ * B, rn(B\) < rR(A) and, therefore, B\<8>RK is a proper non-zero T - invariant subspace of V. This contradiction proves that A/B is an R - periodic module. Let H = CG(A/B). Lemma 8.2 yields that H * < 1 >. Thus //fl P * < 1 > and P/(P fl H) is finite. It follows that Pm < H for some m e N, so that A(aF(Pm)) < B. Then A/B is finite by (2). This contradiction shows that B = < 0 >, i.e. A is a just infinite FG - module. If A is finitely generated by n elements as an R - module , then Ap* is a module over the finite ring F(PIPm)\ clearly Ap« has finite order at most ql where / = n\P : Pm\,q = \F\. It now easily follows that if every finitely generated R submodule of A is contained in a submodule generated by n elements, then Ap* is finite. A derivation 8 : T —<• GLr(K) is said to be irreducible, if K^ has no proper non-zero T - invariant subspace, where the action ofT is specified by the rule vt = v'ts, v e K&, t e T. 8.8. Corollary [RW] The module M(5) is just infinite if and only if 5 is irreducible andM(8)p«' is finite for all m e N. The next natural question is about the isomorphism of M(<5).

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8.9. Theorem [RW] Let F be a finite field, G = P X T be a group satisfying (1) - (3) of Theorem 8.5. The FG - modules M{8) and M(5') are isomorphic if and only ifrR(M5) = rR(MS') as R - modules where R = FP, and ifK is the field of fractions of R, then there exists a matrix © e GLr(K) satisfying the following properties: (l)ts® = ®'ts' for all t e T; (2) the rows of® are in M(8'); (3) the rows of®'1 are in M(5). Proof Assume that a: M(8) —• M(8') is an FG - isomorphism. Then rs(M(5)) = /-fi(M(5'))- Now a extends to a non-singular linear transformation of K^ which can be represented by a matrix © e GLr(K) with respect to the standard basis ofK^; thus aa = a®, a e K^r\ Since (at)a = (aa)t for all t e T, a'ts® = (a®)'ts' = a'®'t5', which shows that ^ 0 = 0 ' ^ . Clearly, the rows of® are in M(8') since they are images under a of the basis vectors. Similarly, the rows of© - 1 are in M(S). Conversely, assume that there is a matrix © e GLr(K) satisfying (1) - (3). The mapping a : a — a®, a e K& is a non-singular linear transformation of K^. The mapping a is an FG isomorphism, as may be seen by reversing the previous arguments. Now R^a < M(8') and R^a~l < M(8) by (2), (3). Hence M(8)a < M{8') and M{8')a~x < M{8) by the definition of M{8). Thus a maps M(8) isomorphically onto M(8'). The following question arises in connection with the results of this chapter and Chapter 6. Let V be a class of all polycyclic-by-finite groups. Question 5 Describe the structure of a V - hypercentral group {even a PC group) G,for which there exists a just infinite DG - module A such that CG(4) = < 1 >. In particular, is Gmetabelian-by-finitel

Chapter 9 Co-Layer-Finite Modules over Dedekind Domains

Let D be a Dedekind domain. AD- module A is called D - co-(layer-finite), or it is simply said that it is co-(layer-fmite) as a D - module, if for every non-zero ideal I of D the factor-module AIAI isfinitelygenerated D-module. In particular, every just infinite DG - module is D - co-{layer- finite). The aim of this chapter is to consider the structure of the co-(layer-finite) Z)-modules; in particular, this study will contribute to the description of just infinite modules, which is indirectly continued here. The results of this chapter are slight extensions of some results of [FdGK 3]. The case D = Z was previously considered in [KL 1]. 9.1. Lemma Let D be a Dedekind domain and let A be a D-module. Then A is co-layer-finite if and only if A/AP" is finitely generated, for any P e Spec{D), andn e N. Proof Suppose that A/AP" is finitely generated, for every P e Spec(D) and every n e N. Given a non-zero ideal /of D, we have I = P\x... P"kk =P"l'f\...nP"k1', where the P, are different non-zero prime ideals of D (see, for example, [KG, Theorem 3.3.3 and Proposition 3.3.9]). Consider the factor-module A = AIAI. Then

101

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A=ATx®-®ATk, where A~p, is the P,-component of A. Since by Lemma 1.3 AjjPi"' = A~pn i ±j, AP"' = ® A~Pj, and, since AlAPj" is finitely generated, we have that APj = AlAPj"' = {AIAr)l{AIAr)Pi"' = (A/AT)/((APj"> +AI)/AI) = (A/AT)/(APj"'/AT) = A/AP,"' is finitely generated, giving that A/AI is finitely generated too. The converse statement is obvious. 9.2. Lemma Let D be a Dedekind domain and let A be a D-module. Then A is D-co-(layer-finite) if and only ifA/Ax isfinitelygenerated, for every 0 * x e D. Proof. The direct assertion is trivial, using Lemma 9.1. For the converse .it suffices to recall that an ideal of a Dedekind ring can be generated by at most two elements (see, for example, [KG, Corollary 3.3.14]). 9.3. Lemma Let D be a Dedekind domain, A a D - module, B a D - submodule ofA. (1) If A is co-(layer-finite), then so isA/B. (2) IfB andA/B are co-(layer-finite), then and A is also co-idayer-finite). Proof (1) is obvious. (2) Let / b e a non-zero ideal of D. Then (A/B)/((AI + B)IB) is finitely generated. Since (AI + B)IAI s B/(B n AT) and

BI<(Bf]AT),

(AI + B)IAI is finitely generated too. Since (A/(AT)/((AI + By AT) = A/(AI + B), A/AI is finitely generated. 9.4. Lemma Let D be a Dedekind domain, A a D-module and B a pure D-submodule ofA. If A is co-(layer-finite), then so is B. Proof

Let 0 * x e D, then B n Ax = Bx, so that B/Bx = BI{B n Ax) = (Ax + B)/Ax < A/Ax.

Since D is a noetherian ring and A/Ax is finitely generated, then and B/Bx is

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103

finitely generated. It suffices to apply Lemma 9.2. 9.5. Lemma Let Dbea Dedekind domain, P e Spec(D), A a co-(layer-finite) D-module. If A is a P-module, then there exists a direct decomposition A = B@C, where B is finitely generated and C is D-divisible. Indeed, since AIAP is finitely generated, dimoip{AIAP) is finite. Now we may apply Corollary 5.4. 9.6. Corollary Let D be a Dedekind domain and let Abe a periodic D-module. If A a co-(layer-finite), then A

= ®PznD(A)Ap>

where Ap = Bp © Cp is the P-component ofA, Bp is finitely generated and Cp is divisible. 9.7. Lemma Let D be a Dedekind domain and let A be a torsion-free D-module having an ascending chain ofD-pure submodules <0> = A0

such that roiAi+i/Ai) = I, for every i e N. Then A is co-(layer-finite) if and only if for every P e Spec(D) there exists a number h(P) e N such that (A„+l/A„)P = A„+i/A„for each n > h{P). Proof Suppose that A is co-(layer-finite) and assume that there exists P e Spec{D) such that (A„+\/A„)P * An+\IA„, for infinitely many numbers n. Then ,4 has an ascending chain of D-pure submodules < 0 > = Bo < B\ < ... < B„ < B„+i < ... such that rD{Bi+\IBi) is finite and (5,+i/B,)P * 5,+i/B, for every j e N, and A = U«eN^"- I n particular, B\P * B\. Consider the factor module BilB\P. Clearly B\IB\P is the D - periodic part ofB2fBiP and, since AnnD{B\IB\P) = P, we may decompose B2IB\P = {B\IBXP) © (C/BiP) [KI], where C is some D submodule. Then CIB\P = BilBy and so {CIB\P)P * (OBi)P. From the equality (B2/BiP)P = {CIBiP)P = CP/(Bi)P it follows, that dimDiP(B2IB2P) > 2, and it is very easy to establish inductively that dimD/P(B„/B„P) > n, for every « e N. Moreover, A = \JneNB„ gives that AP = \J„eNB„P, and so A/AP = \JneN((Bn+APyAP). Since AIB„ is torsion-free, then B„IB„P is the periodic part of AIB„P, and again we obtain a decomposition AIB„P = (B„/B„P) © (E/B„P), for certain D -

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submodule E. It follows that (A/B„P)P = (E/B„P)P = EPIBnP < EIB„P. In particular, AP f| B„ = B„P, and hence (B„+AP)/AP

= S„/(5„ n ^ P ) = 5„/£„P,

so that dimD/P(B„ +APIAP) > n, for each « e N. This means that dimD/F(A/AP) is infinite, a contradiction. Conversely, suppose that the jD-module A satisfies the conditions in the statement. For any P e Spec(D), we put h = h(P) It follows that (A„+hIAh)P = A„+fl/Ah, for every n e N. Consider the factor-module A„+/,/A/,)P; obviously Ah/AhP is its periodic part. Since Anno(Ah/AhP) = P, again A„+hIAhP = (Ah/AhP) © {Cn+hlAhP) for some D - submodule Cn+h. Since Cn+h/AhP = An+h/Ah, (C„+hIAhP)P = C„+h/AhP. Thus (A„+h/AhP)P = A„+hPIAkP = C„+f,/AhP, which allows us to check that ^4„+>,A4 „+/,/> = Ah/AhP. Hence dimD/p(A„+h/An+hP) = dimD/p(Ah/AhP)

= /w.

Again/f = L U N ^ gives that^A4P = L L N ^ " ^ + AP)IAP. Since (A„+h +AP)IAP = An+hf(An+h HAP) = A„+h/A„+hP, we have that dimD/p((An+h+AP)/AP) = m, for every n e R Since A„+h < A„+h+U (,A„+h +AP)/AP = (An+h+i +AP)/AP, A/AP = (Ah+AP)/AP, and hence dimD/P(A/AP) = m is finite. By Corollary 5.23, the module ^A4/" is finitely generated, for any t e N. Thus, Lemma 9.1 gives that A is co-(layer-finite), as required. 9.8. Lemma Let D be a Dedekind domain such that Spec{D) is countable and let A be a co-(lqyer-finite) torsion-free D-module. Then A includes a D-pure submodule B such that ro(B) is countable andAlB is divisible. Proof Put Spec(D) = {P„ submodule B, for every B\ - © A S A C*> where Ci example, [PD 2, Theorem © A E A , CxJ&Pi.

| n e N>. By Theorem 5.18, A includes a 7\-basic /' e N. Since B\ is a projective D-module, are isomorphic to certain ideals h of D (see, for 7.7]). Then 5 i P i = © ^ C^P and BJBxPx =

Further, C V C A P I = V / A P I = D/Pi (see, for example, [KG,

Corollary 3.3.15]). From Bi f)AP\ = B\P\ we obtain that Bi/BiPi

= Bi/(Bi r\APi)

= (5,

+APi)/APi.

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105

It follows that B\IB\P\ is finitely generated. In other words dimD/Pl(B\/B\P) is finite and, since dimDiP^{B\IB\P) = |Ai|, we conclude that Ai is a finite set and hence B\ is finitely generated too. LetAi/Bi be the periodic part of AIB\\ in particular, ro(A\) = ro(B\), so that A\ has finite D-rank. Moreover, AIA\ is a torsion-free D-module and, since (AIB\)PX = AIB\, we have that (AIA\)P\ = AIA\. Repeating these arguments, we construct an ascending chain of D-pure submodules A\ < ...*'. Then G4/fl)x = {AIB){xD) = Ol/flXP*1 .../>*', so that v4/2? is divisible. It is worth mentioning that among the Dedekind domains, which satisfy the above hypothesis are the ring Z of ordinary integers and any group-ring of the form Wp < x >, p is a prime. As it is well known, these rings have many important group-theoretical applications. 9.9. Corollary Let D be a Dedekind domain such that Spec{D) is countable and let A be a torsion-free D-module. Then A is co-{layer-finite) if and only ifA has an ascending chain <0> = A0

= \J„£NA„
=A

ofD-pure submodules such that: (i) A/A m is divisible; (ii)rD(An+\IA„) = I, for every n e N; (iii)/or each P e Spec{D) there exists a number h(P) e N such that (A„+i/A„)P =A„+UA„foreachn> h(P). Proof Suppose first that A is co-layer-finite. By Lemma 9.8, there exists a D-pure submodule of countable D-rank such that AIB is divisible. By Proposition 5.15 5 has an ascending chain of D-pure submodules < 0 > = A0
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co-(layer-finite). Then Lemma 9.7 shows that for every P e Spec{D) there exists a number h(P) e N such that (A„+i/A„)P = A„+1/A„ for each n > h{P). Conversely, if A satisfies the above conditions, then by Lemma 9.7 the £>-submodulev4m is co-(layer-finite). Obviously, A/A a is co-(layer-finite), since it is divisible. Then A is also co-(layer-finite) by Lemma 9.3. 9.10.Theorem Let D be a Dedekind domain such that Spec(D) is countable and let A be a D-module. Then A is co-layer-finite if and only ifA decomposes as A = B®E and (i) D - submodule E is divisible and periodic; (ii) the submodule B has an ascending chain ofD-pure submodules 0 = B0 < Bi < ... < B„ < ...Ba < Bm+l = B such that B\ is the D - periodic part ofB, BIBm is divisible andro(Bn+ilBn) = 1, for every n e N; (Hi) for each P e Spec(D) there exists a number h(P) e N such that (B„+l/B„)P = Bn+X/B„ for each n> h(P); (iv) B\ = ® P e n ,B j Cp, where Cp is the P-component ofB\, being each Cp finitely generated. Proof Suppose first that A is co-layer-finite. If T is the D - periodic part of A, by Corollary 9.6 and Lemma 9.4, we may express T = (BpenD(A) ^p' w n e r e Tp = Cp® Ep is the P-component of A, Cp is finitely generated and Ep is divisible. Define E = ®Pen rA\ Ep, so that E is divisible and henceforth a direct summand of A : A = E © B, for some B (see, for example, [SV, Proposition 2.10 and Theorem 2.15]). Consider now B\ = ®P^UDIA) Cp; clearly B\ is the D - periodic part of B. Thus BIB\ is torsion-free and it suffices to apply Corollary 9.9. Conversely, by Corollary 9.9, BIB\ is co-(layer-finite). Clearly, so are B\ and E. Then it suffices to apply Lemma 9.3.

Part III Just non-A' -Groups

The set of all proper factor - groups of a group (i.e. the factor-groups by non-identity normal subgroups) is the widest family of group quotients. Let X be a class of groups. A group G is said to be a just non-X - group if every proper factor-group of G is an X - group but G does not belong to X. This part of the book is dedicated to just non-A" -groups for such natural classes of groups X as abelian groups, finite groups and some their extensions, nilpotent and hypercentral groups, central-by-finite groups, groups with finite derived subgroups, FC - groups, CC - groups. We will study such groups G with some additional conditions, which imply the non-simplicity of G. The following condition is typical for these purposes: Fitt{G) * < 1 >, where Fitt{G) is the Fitting subgroup of G, that is the subgroup generated by all normal nilpotent subgroups. Naturally, the first step here is investigation of a structure of the Fitting subgroup of just non-A" -groups. In fact, for groups from all indicated classes it is possible to prove that its Fitting subgroups are abelian. This allows us to consider them as modules over the group GIFitt{G) already belonging to the class X. The subsequent study is divided into two cases: the monolithic case and the non-monolithic case. In the monolithic case, as a rule, it is possible to prove, that Fitt(G) coincides with the monolith of a group; in the non-monolithic case Fitt{G) stands by a just infinite module. Therefore a tool in this study is the developing of properties of simple and just infinite modules, which were investigated in the previous chapters.

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Chapter 10 The Fitting Subgroup of Some Just non- X -Groups

In this chapter, we collect some of the most relevant properties of just non-X -groups we need in the sequel, especially those which concern the Fitting subgroup. Let G be a group. We recall that G is said to be monolithic if the intersection M of all its non-identity normal subgroups is non-identity: in this event M is called the monolith of G and will be denoted n(G). Evidently p.(G) is the unique minimal normal subgroup of G. Otherwise, if that intersection is identity, G is said to be non-monolithic. Clearly we have 10.1. Lemma Let X be a class of groups which is closed under taking subgroups and Cartesian products, that is, X is S and C-closed, (in particular, if Xis a variety ofgroups [RD 9]). Then a just non-X -group G is monolithic. Proof If S is the set of all non-identity normal subgroups of G and M = C\S, by Remak's theorem we can embed GIM'm the Cartesian product ri// e ,s GIH. Since G is a just non-A1-group and X is S and C-closed, it follows that GIM e X. But G <£ X, so that M * < 1 >. 10.2. Corollary (i) A just non-abelian group is monolithic. (ii) IfNc denotes the class of all nilpotent groups ofnilpotency class < c, then a just non-Nc-group is monolithic. 10.3. Lemma

Let X be a formation of groups and let G be a just non-X-group.

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110

Then G does not include two non-identity normal subgroups U and V such that

un v = < 1 >. Proof

Proceed as in the proof of Lemma 10.1.

10.4. Lemma [KO 2] Let G be a just non-FC-hypercentral nilpotent normal subgroup ofG, then L is abelian.

group. If L is a

Proof Suppose that L is non-abelian and choose in L a maximal G-invariant abelian subgroup A; thus A * L. Put ZIA = t^(LIA). Then ZIA is a non-identity G-invariant subgroup of LI A and ZIA f)FC(G/A) * < 1 > by Lemma 3.3. Let A * x0A e ZIA n FC(G/A) and define XIA = (< x0 >a )AIA. Then XIA is finitely generated abelian and the index \GIA : CGIA{XIA)\ is finite. Put YIA = CGIA(XIA). By the choice of A, X is non-abelian. Let x e X and consider the mapping
= [^.JCW] = [ay ,u\[ay ,x]u = [ay,u] = (<*>>*.

Actually p* is a ZK-endomorphism of A. Put Zi = f(L), so that Zi * < 1 >, and hence Zi < ^ . Obviously, J7Zi is FC-hypercentral and, since Z\ < CA(X), then YICA(X) is also FC-hypercentral. Suppose AICA(X) * < 1 >. Besides, Corollary 3.4 implies that A/CA(X) (\FC(YICA(X) *< 1 >. Let CA (A) * a d (A) e ^ / Q ( A ) 0 FC(YICA(X) and choose /7C^(i0 e Cy / C / 4 W (aQ(A)). Now aA = aft, where ft e C,i(X), and {a
conclude has to be C^(A), a —• [c,x], = (cy)9x, cv where

(c6x)y = ( c ^ e , = (cv)0, = (cfl,)(v0 x ) = c0x. This means that / m ^ < £(10- Since Xcannot be abelian, there exists some x e X

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111

such that ImOx * < 1 >. Thus £(F) * < 1 > and, since \G : Y\ is finite, we arrive again to FC(G) * < 1 >. Hence L is abelian. 10.5. Theorem [KO 2] Let G be a just non-FC-hypercentral group, F = Fitt(G) * < 1 >.Then F is abelian and either F is torsion-free or there exists a prime p such that F is p-elementary abelian. Furthermore, CQ (F) = F. Proof Let x,y e F, then there are the nilpotent normal subgroups Lx, Ly such that x e Lx, y e Ly. By the Fitting's theorem (see, for example, [RD 9, Theorem 2.18]), LxLy is nilpotent and so abelian by Lemma 10.4. Thus F is abelian. Let Tbe the periodic part of F. If T * < 1 >, then by Lemma 10.2 there exists a prime p such that T is a p-group. Suppose that 1? * < 1 >. Then T* T\ =Qi(T) = {x e T \ xP = 1>. Since GIT\ is FC-hypercentral, then by Corollary 3.4 (T/T{) f] FC(G/Ti) * < 1 >. In particular, TIT\ includes a finite non-identity G-invariant subgroup PIT\. Put HIT\ = CciT\(PIT\). Then H is a normal subgroup of finite index. If h e H and c € P\T\, then Ti = [cTuhTi] = [c,/j]ri and so [c,h] eTi.lt follows that [C^/J] = [c,K\p = 1Since c £ T\,cp * 1, and so £(//) * < 1 >. As other times, this yields that FC(G) * < 1 >, because \G : H\ is finite. This contradiction shows that T^ = < 1 >, that is, T = T\ is elementary abelian. Suppose now that F ± T. Then we may decompose F = A x r for some subgroup ^ (see, for example, [FL 1, Theorem 27.5]). It follows that Fp < A and, in particular, Fp fl T = < 1 >. Since F ± T,FP * < 1 >. This contradicts Lemma 10.2. Thus F = Tas claimed. For the last assertion, suppose that C = CG{F) * F. Since GIF is FC-hypercentral, by Corollary 3.4 we have that (OF) n FC{GIF) * 1. Let F*xFe (OF) n FC(GIF) and define XIF =< xF >G/F . Then XIF is central-by-finite. If RIF = C,(XIF), then 7? is a nilpotent normal subgroup of G, so i? < F. It follows that XIF is finite. Since F < ((X), by Schur's theorem (see, for example, [RD 9, Theorem 4.12]) the derived subgroup [X,X\ is finite. Therefore [X,X\ = < 1 >, andXbecomes abelian. Hence X < F, which is a contradiction. We immediately obtain the following consequences. 10.6.Corollary [KO 2] Let G be a just non-CC-group. IfFC(G) = < 1 > and Fitt(G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Furthermore, CG(Fitt(G)) = Fitt(G). 10.7. Corollary [FdeGK 3] Let G be a just non-FC-group. If FC(G) = < 1 > and Fitt(G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup.

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Moreover, CG{Fitt{G))

= Fitt{G).

In considering of just non-hypercentral groups, we obtain similar results. For the proof we shall need other auxiliary statements. 10.8. Lemma [KSU 3] Let G be an infinite just non-hypercentral-group. includes a finite normal subgroup, then G is finite.

If G

Proof Suppose G is infinite. Let A be a finite minimal normal subgroup of G. If A is not abelian, then A % CG(4) and so A n CG(A) = < 1 >. But, since G is infinite and G/CG(A) is finite, we have that CG(A) * < 1 >, which contradicts Lemma 10.3. Therefore A is abelian. Indeed, by minimality, A is elementary abelian. Since GIA is hypercentral but G is not, it turns out that A is exactly the hypercentral residual of G. By Robinson's Theorem 4.5, G splits over A, that is G = A X H for some subgroup H. Obviously this H is an infinite hypercentral subgroup and \H : CH{A)\ is infinite; in particular CH(A) * < 1 >. Since CH(A) is normal in H, C = CH{A) (1 £(//) * < 1 >• But C < C(G), so C(G) * < 1 >, a contradiction. 10.9. Theorem [KSU 3] Let G be an infinite just non-hypercentral group. If Fitt{G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Moreover CG{Fitt{G)) = Fitt(G). Proof Suppose that FC{G) * < 1 >, pick l * j e FC{G) an put X = < x >G . If \x\ is finite, then X is finite too. Thus Lemma 10.8 shows that |x| is infinite. It follows that X includes a non-identity G-invariant torsion-free abelian subgroup A. Let r = ro(A). Given a prime p, we have that Ap * < 1 > and GIAP is hypercentral. Since \GIAP\ = pr, GIAP < (,r(GIAp). In other words [Gr,A] = [G,...,G,A] <AP. Since this holds for every prime p, r

\Gr,A\ < pi e¥Ap = < 1 >, and so A < £V(G) and gives G hypercentral. This contradiction shows that FC(G) =< 1 >, and then it suffices to apply Theorem 10.5. 10.10. Theorem [RW] Let G be a just non-(polycyclic-by-finite)-group. IfFitt G * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Moreover CG(Fitt(G)) = Fitt(G). Proof Since G satisfies the maximal condition on normal subgroups, A = Fitt{G) is nilpotent. Suppose that B = [A,A] * < 1 >. Then GIB is polycyclic-by-finite and so AIB is finitely generated. Moreover, by a result due to

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113

Baer, the group A itself is finitely generated (see [RD 9, Corollary of Theorem 2.26]). Therefore G is polycyclic-by-finite, a contradiction which shows that A has to be abelian. Let Tbe the periodic part of A and suppose that T * < 1 >. By Lemma 10.3, there exists a prime p such that T is a p-group. Put T\ = Qi(T) so that T\ is an (infinite) elementary abelian p-group and GIT\ is polycyclic-by-finite. It follows that TIT\ is finite and in particular T is bounded. A celebrated result due to Priifer allows us to express T as a direct product of cyclic groups T =XASA < t% > (see, for example, [FL 1, Theorem 17.2]). Thus, if T * Tu then 1? * < 1 > and TIV is finite. Since V = T/Qi(T) is finite, we can deduce that Tis finite. Since this is impossible, T = T\. Suppose that T 4= A. We may decompose A = Tx U, for some subgroup U (see, for example, [FL 1, Theorem 27.5]). It follows that Ap < U, and, in particular, Ap is a non-identity torsion-free normal subgroup. Therefore Ap C\ T = < 1 >, which contradicts Lemma 10.3. This shows thaty4 = T. Finally, the assertion CG(A) = A can be proved proceeding as in the proof of Theorem 10.5.

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Chapter 11 Just non-Abelian Groups

Chronologically the class of just non-abelian groups was the first class of just non-A'-groups. The structure of soluble such groups was determined by M. F. Newman in his papers [NM 1] and [NM 2]. Making use of the results previously showed in the precedent chapters of this book, we can carry out this study more concisely. Let G be a just non-abelian group; by Corollary 10.2 G is monolithic. Thus let M = n(G) be the monolith of G. As mentioned above, we suppose that Fitt(G) * < 1 >. If M is not central in G, then £(G) = < 1 >, but M is abelian. Let A be a maximal abelian normal subgroup of G. Considering A as a Z//-module, where H = GIA is an abelian group, we can think of Mas a simple Z//-submodule of A. Suppose that M * A. If AIM has elements of finite order, we may choose in AIM a cyclic subgroup CIM of prime order. Note that C is a Z//-module since GIM is abelian, and so CIM is a simple Z//-module as well. By Corollary 4.3, there exists a non-identity G - invariant subgroup E < C such that C = M x E. In particular, MC\E =< 1 >. This contradicts Lemma 10.3 and so AIM is torsion-free. Suppose that M is an elementary abelian p-group, for some prime p. Then there exists a subgroup U such that A = Mx U (see, for example, [FL 1, Theorem 27.5]). Therefore < 1 > * Ap < U and so Ap C\ M = < 1 >, which contradicts Lemma 10.3. However, if M is torsion-free, then Corollary 4.4 and again Lemma 10.3 lead to another contradiction. Consequently A = M and so the monolith of G is a maximal abelian normal subgroup of G. Let L be a nilpotent normal subgroup of G; then f (I) = M. Suppose that L * M and choose an element x e DM, we can form the abelian group K = < x, M >. This is a normal subgroup of G since GIM is abelian. However M * K, which is impossible. Thus M = L and hence M = Fitt(G). We may apply the Robinson's result (Theorem 4.5) to obtain that G conjugately splits over M. On the other hand, if S is a complement to M in G, S = GIM is abelian and we may think of M as a simple ZS-module and apply

115

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Just non-X-Groups

Corollary 2.4 and Theorem 2.6. All this gives the case in which the monolith is not central, which is our next result. 11.1. Theorem [NM 1] Let G be a just non-abelian group with Fitt(G) * < 1 > and ((G) = < 1 >. Let M be the monolith ofG. Then: (1) M - Fitt(G) = CG(M) is the unique maximal abelian normal subgroup of G. (2)M=[G,G\. (3) G = M X S, S is abelian and any complement to M in G is conjugate to S. (4) The periodic part T = t(S) ofS is a locally cyclic group. (5) IfM is an elementary abelian p - subgroup for some prime p, then T is a pi - subgroup. (6) Ifro(G/M) is finite, then G is periodic. Obviously, these conditions above are also sufficient. Suppose now that the monolith M = ji(G) is central in G. In particular, ((G) * < 1 > and then G is nilpotent. Obviously M has to be cyclic of order p, for some prime p, and besides M = [G,G]. For each g e G, the mapping

which shows that GI((G) is an elementary abelian/?-group. By Lemma 10.3, ((G) is a/?-group and its lower layer £l\(((G)) = M. It follows that either ((G) is a Prufer/7-group or ((G) is a cyclic p-group. Put C = ((G); it is easy to decompose GIM= C/MxE/M for some E, and then we may express G = CE and Cp\E = M. This means that [E,E] = ((E) = fi(G) and El((E) is an elementary abelian p - group. In other words, E is an extraspecial p-group. Conversely, suppose we are given ap-group C,p& prime, which is either a Priifer group or a cyclic group, and an extraspecial p-group E; let < e > = [E,E] = ((E) and < c > = Qi (C). Define H= ExC and L=< ce'x >. Then G = GIL = EC where E = ELIL s EI(E DL) = E, C= CLIL = CI(C f]L) = C, and E f] C = < e >< ((G), where e = eL, an element of order p. Clearly, ((G) = ((E)C = C, and [G,G] =< e >. If U is a non-identity normal

Just non-Abelian Groups

117

subgroup of G, then U (~l C(G) * < 1 >, because G is nilpotent. Since £(G) = C is a locally cyclic /? - subgroup and < e_> = Qi(C), then [G,G] = < e > < U. It follows that GIU is abelian and hence G is a just non-abelian group. So we have just obtained the following result. 11.2. Theorem [NM 2] Let E be an extraspecialp-group and C a locally cyclic p-group. Suppose that < e > = [E,E] = £(£), < c> = Qi(C). Define H = ExC andL =< ce _1 >. ThenG = HIL is a just non-abelian group and fi(G) < (,{G). Conversely, every just non-abelian group G with f(G) * < 1 > can be obtained in this way. This last result raises the question about studying extraspecial p-groups if one wants to know more on (nilpotent) just non-abelian groups. The remainder of this chapter is dedicated to the description of some details of extraspecial p-groups. Let p be a prime and E an extraspecial p-group. Then A = E/£(E) is an elementary abelian p-group and can be thought as a vector space over the prime field Fp. Put < e > = [E,E] = £(£), a subgroup of order p. If a = aC,(E) and b = b^(E), then [a, b] = ek, for some k e Z; this k is unique modulo p and we indicate this writing k e Fp. If we change the representatives a = a\Q{E) and b = b\£(E), then we may write a\ = ac\, b\ = bci, where c\,c2 e £(£). Operating, we have that [a\,b\] = [aci,bc2] = [aci,6][aci,c 2 ] = [ac\,b] = [a,b][c\,b] = [a,b], which shows that we may define correctly the mapping is a bilinear mapping. In fact, given c = cC,{E), if we suppose that [a,c] = ek', [b,c] = e*2, then we have O M = [a,c][b,c] = e*'e*2 = e*,+*2, which means that cp(a + i,c) = k\ + k2 = is non-degenerate, because £(£) = < e >. Thus, an extraspecial p-group gives rise to a non-degenerate symplectic space. Conversely, let A be a non-degenerate symplectic space over the field F and let (p:AxA - » F - its corresponding alternate bilinear form. Consider the set E= {{a,a)\a e A,a e F}

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and define a multiplication on £ by the rule (a,a)(b,B) = (a + b,a + B + q>{a,b)). One easily checks that this makes the set E into a group in which the pair (0,0) is the identity element and (a,a)~l = (-a,-a). Let C = {(0,a)|a e F}; then C is a subgroup of E which is isomorphic to the additive group of F. Since [(a,a),(b,p)]

= (0,2p(a,6)),

we may deduce that [(a, a), (0, /})] = (0,0), and all this shows that [£,£]< C < £(£). Since q> is non-degenerate, for every 0 * a e A there is some * e A with ) and their corresponding notions in E. If a\,ai e A, then a\,ai are said to be orthogonal if F- be its corresponding bilinear form. IfdintFA is countable, then A is an orthogonal direct sum of hyperbolic planes.

Just non-Abelian Groups

119

Proof Let {a„ \ n e N} be a basis of A. Since ^ is a non-degenerate space, there is an index ;' e N such that 2. If (e2,a\„) = 0. Consider the subspace ^4„ = a\F+ ... + a„F. Since the matrix

f

< 0

H P*

0

0 0 .. 0 0 ^

1 -C<3

0 0 .. 0 0 1 0 .. 0 0

-a4

0

1 .. 0 0

0

0

V P»

0 1

J

is non-singular, then {e\,ei,a\-$,... ,a\„} is a basis of A„. Since this is valid for every n > 2, then {e\,e2,ct\n | « > 2} is a basis of A. Put H\ = e\F+e2F; clearly H\ is a hyperbolic plane, and H\x = {a\„ \ n > 2}. Moreover, A=Hx+H\

=

Hi&Bu

where B\ = {a\„ \ n > 2}. In the same way, we decompose B | in a orthogonal direct sum of another hyperbolic plane H2 and another subspace B2. Applying induction, we may construct a countably infinite sequence of hyperbolic planes H\,H2,... ,H„,... such that

as required. 11.4. Theorem A countably infinite extraspecial p-group, p a prime, is a direct product of {non-abelian) groups of order p3 with their centers amalgamated. Proof We apply all the above ideas. Given the countably infinite extraspecial jP-group E, if C = C(£)> w e m a y transform A = EIC into a non-degenerate symplectic space over ¥p of countable dimension. By Proposition 10.3, A

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decomposes into an orthogonal direct sum of hyperbolic planes and the pre-image of each one of these hyperbolic planes corresponds to non-abelian subgroups of £ of order p3, which cuts each to other in C. As E is generated by all these pre-images, then the result follows. It is worth mentioning that, for uncountable groups, we have no analogy of Theorem 10.4 as the following example shows. 11.5. Example [HP 3] Given a prime p, let < a >,< b„ >,< c„ >,n e N, copies of the cyclic group of order p. Form B =X„<=N < b„ > and C = n „ e N < c» >- Then C acts on the direct product < a > xB by the rules given by ac» = a,bc„" = b„a,bcn" = b „ , n,k e N , n =f= k.

Let E be the corresponding semidirect product. By construction £(£) = [£,£] = < a >. Since E/£(E) is an elementary abelian p-group, then E becomes an extraspecial p-group. Suppose we decompose £ in a direct products of groups Hi, A e A, of groups of order/?3 with < a > amalgamated. Since E is uncountable, so is A. However B is countable so that there exists a countable subset T c A such that B < Er =< H\ \ A e T >. This gives that < Hx \ A e AT > < CE(B) and, in particular, EICE{B) is countable. But CE(B) = < a > x B and £/(< a > xB) = C is uncountable. This contradiction shows that our assumption is not possible. The structure of uncountable extraspecial p-groups is very complicated and remains still almost unknown. A survey of the known results is reported in [TM 2, Section 3].

Chapter 12 Just non-Hypercentral Groups and Just non-Hypercentral Modules

The class of all nilpotent groups and the class of all hypercentral groups are very natural extensions of the class of all abelian groups. Thus in our study of just non-A'-groups, the consideration of just non-hypercentral groups and just non-nilpotent groups should be the next step. As in the previous chapters we are studying the just non-hypercentral groups G with Fitt(G) * < 1 > . Some information about this subgroup was obtained in Chapter 10. In particular, A = Fitt(G) is abelian, and, as usually, we can consider A as a ZH - module, where H = GIA is a hypercentral group. If £ is a non-identity G - invariant subgroup( or, using the module language, B is a non-zero ZH - submodule) of A, then GIB is hypercentral, in particular, a ZH -factor-module AIB is ZH hypercentral. Let R be a ring, G a group. An RG - module A is said to be a just non-hypercentral (more precisely, just non-RG-hypercentral) if every proper factor-module of A is RG - hypercentral, but A is not RG - hypercentral. Similarly, A is said to be just non-nilpotent, if A is not RG- nilpotent, but every proper factor-module ofA is RG - nilpotent. Thus, the study of just non-hypercentral groups naturally requires the necessity of study of just non-hypercentral modules. Note that the study of these modules leads us to some other results 12.1. Lemma Let R be a ring, G a group, A a just non-RG-hypercentral module, (p an non-zero RG - endomorphism ofA. Then

are the R submodules of A. If we suppose that Ker

, then AIKer

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hypercentral. However Im

, then A/Imcp is RG - hypercentral. However, in this case the module A is RG hypercentral. It follows that Im(p = < 0 >, that is q> = 0. This means that Ker

. 12.2. Corollary Let R be a ring, G a group, A a just non-RG-hypercentral module. Then EndRGiA) has no zero-divisors. 12.3. Corollary Let D be a Dedekind domain, G a group, A a just non-DG-hypercentral module, I = AnnoG(A), CII the center of the factor-ring DGII. Then CII is an integral domain. Proof For each element x e C the mapping ix : a —• ax, a e A, is a DG endomorphism of A. Furthermore, the mapping <E>: x —• ix,x e C, obviously, it is a homomorphism of the ring C in the ring EndoaiA) and Kerd? = Annix}{A) = I. By Corollary 12.2 C//is an integral domain. 12.4. Corollary Let D be a Dedekind domain, G a group and A a just non-hypercentral DG - module. Then either A is D - torsion-free or AnnoiA) = P e Spec(D). Clearly, D < C, therefore DI{Dr\Annoa{A)) is an integral domain. It follows that either (D f] AnnDG{A)) = < 0 > or (D fl AnnoG(.A)) is a maximal ideal of D. 12.5. Corollary Let R be a ring, G a group, l * z e C(G), A a just non-hypercentral RG - module, CG(A) = < 1 >. Then CA(Z) = < 0 >. Proof

Since z e £(G), the mapping (p : a —• a(z- 1), a e A,

is an RG - endomorphism of A. By Lemma 12.1 Kercp - Ann^{z - 1) = CA(Z) = < 0 >. 12.6. Lemma Let R be a ring, G a group, A an RG- module, U an upper RG hypercenter ofA. IfB is a non-zero submodule ofU, then B C\ £RG(A) * < 0 >. This statement could be proved precisely in the same way as its group-theoretic analogy. Denote by IIRG(A) the RG - monolith of module A, that is the intersection of all non-zero submodules of A. An RG - module A is said to be an RG - monolithic, if JJ.RG{A) * < 0 >, and

Just non-Hypercentral Groups and Just non-Hypercentral Modules

non-monolithic

123

otherwise.

12.7. Lemma Let Rbe a ring, G a hypercentral group, A an RG - module. If A is just non-hypercentral andRG - monolithic, then A is a simple RG - module. Proof Let M = HRG(A); then M =£< 0 > and M is a simple RG - submodule. Furthermore, £RG(A) does not include M. Suppose that A *• M. A factor-module AIM is RG - hypercentral, so that £RQ{AIM) = CIM * < 0 >. We can assume that CQ(M) = < 1 >. Let 1 * z e f(G), then the mapping q> : c —* c(z - l ) , c e C, is an RG- endomorphism of C, so that Imcp = C(z - 1) and Ker

, that is C = Imcp. Since C(z - 1) < M, Im

. By Theorem 10.9 A is abelian. Lemma 12.7 yields that A is a minimal normal abelian subgroup, that is A = M. The same Theorem 10.9 proves that CG(.M) = M. Since Fitt(G) includes every normal abelian subgroup, then M i s also a maximal normal abelian subgroup of G. 12.9. Lemma Let F be afield, G a hypercentral group, x an element of infinite order of the center C,(G), A an FG - module such that CG(A) = < 1 >. If A is just non-FG-hypercentral and non-monolithic, then A is F < x > - torsion-free. Proof Put J = F < x >, then J is a principal ideal domain. We will consider A as JG - module. Suppose that A is not J - torsion-free. By Corollary 12.4 Annj(A) = P e Spec(f). There exists an irreducible polynomial J[x) e J such that P = Jf{x). Since C,JG{A) =< 0 >, f(x)±{x-\), in particular, J{x - 1) + Jf(x) = J. Since A is non-monolithic, it includes a proper non-zero FGsubmodule B. Since AIB is FG -hypercentral, CIB = C,FG{AIB) *< 0 >, that is C * B. If c e OB, then c(x - 1) e B. On the other hand, cj%x) = 0. The equation J(x - 1) + Jf{x) = J implies that c + B = B, a contradiction. This contradiction shows XhaXA is J - torsion-free. 12.10. Lemma Let F be afield, G a hypercentral group, A an FG - module such that CG{A) = < 1 >. If A is just non-FG-hypercentral and non-monolithic, then G is torsion-free. Proof Let T be the periodic part of G and suppose that T * < 1 >. Assume firstly that charF = p > 0. Let Tp be a Sylow p- subgroup of G. Suppose that

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Tp *< 1 >; then Tp n C(G) * < 1 >. Choose an element 1 * x e 7> n £(G). Since char F = p, the additive group ^ is an elementary abelian p - subgroup. If follows that the natural semidirect product A X< x > is nilpotent ( see, for example, [RD 10, Lemma 6.34]), in particular, CA(X) * < 0 >. This contradicts Corollary 12.5. Therefore T is a pi- subgroup. Choose a non-identity element y e C, (G) fl T. Since A is non-monolithic, it includes a proper non-zero FG submodule B. Then AIB is FG - hypercentral, so £FG(A/B) = OB * < 0 >. By Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]) there is an F < y > - submodule E such that C = E © B, in particular, E(y - 1) < £. On the other hand, C(y - 1) < B, so that £(y - 1) < 5. It follows that E(y - 1) = < 0 >, that is E < CAM- However, this contradicts Corollary 12.5. If char F = 0, then we repeat the arguments of the previous paragraph, and obtain again a contradiction, which shows that T = < 0 >. 12.11. Corollary Let G be a just non-hypercentral group, Fitt(G) * < 1 >. Suppose that Fitt{G) is an elementary abelian p - subgroup for some prime p. Then GIFitt{G) is torsion-free. 12.12. Theorem Let F be afield, G a hypercentral group of finite 0 - rank, A a non-monolithic FG - module. If every proper factor-module of A is FG hypercentral, then A is itselfFG -hypercentral. Proof Suppose the contrary. Then A is just non-FG-hypercentral. We can assume that Co(A) = < 1 >. By Lemma 12.10 G is torsion-free. Choose a non-identity element x e £(G). Put J = F < x >, then J is a principal ideal domain with the infinite set Spec(J). By Lemma 12.9 A is J - torsion-free. Let 0 * b € A, B = bFG, n= {P \ P e Spec(J) and BP * B}. By Theorem 1.15 the set n is infinite. In particular, we can find a maximal ideal Pen such that P * J(x - 1). Since J is a principal ideal domain, there is an element y e J such that P = Jy. There are the elements u,v e J such that 1 = yu + (x - l)v. Since A is J - torsion-free, BP * < 0 >, thus AIBP is a FG - hypercentral module. By Lemma 12.6 CIBP = BIBP f) C, FG(A/BP) * < 0 >. For every element c e C\BP we have c(x-l)eBP. On the other hand, cy e BP, hence c + BP = cl + BP = cyu + c(x - l)v + BP = BP. This contradiction proves that A is an FG - hypercentral module. 12.13. Lemma Let R be a ring, G a group, A a just non-RG-hypercentral RG module. IfB, C are two non-zero RG - submodule , then BDC *< 0 >. Indeed, if we suppose that B f] C =< 0 >, then using Remak's theorem, we obtain the embedding, A < AIB © AIC, which proves that .4 is RG - hypercentral. Let D be a Dedekind domain, G a group, A a DG- module. Suppose that A is D

Just non-Hypercentral Groups and Just non-Hypercentral Modules

125

- torsion-free. Put V = {B | B is a non-zero pure DG - submodule of A}, PDG(A) =

f)V

12.14. Lemma Let D be a Dedekind domain, G a hypercentral group, A a DG module which is D - torsion - free. Suppose that A is a just non-hypercentral module andPDG{A) * < 0 >. Then PDG(A) = A. Proof Suppose the contrary, let R = PDG{A) * A. Then AIR is D - torsion free. Furthermore, clearly R is a DG - submodule, so that AIR is .DG-hypercentral. Hence CIR = C,DG{AIR) * < 0 >. Let B be a non-zero DG - submodule of R. It follows from the choice of R that RIB is D - periodic. In other words, R is D irreducible. Choose an element c e C\R, then (c + R)DG = cD + R. Since A is just non-hypercentral, G * CG(R)- Corollary 4.4 shows that C includes a non-zero DG - submodule E such that E n R = < 0 >. But this contradicts Lemma 12.13. This contradiction proves the equality A = PDG(A). 12.15. Lemma Let D be a Dedekind domain, G a hypercentral group of finite 0 - rank, A a non-monolithic DG - module which is D - torsion -free. Suppose that A is just non-hypercentral module and the set Spec(D) is infinite. Then PDG(4) = <0>. Proof Suppose that PDG(4) * < 0 >. Lemma 12.14 yields that ,4 = PDG(A). In other words, for every non-zero DG - submodule B of A the factor-module AIB is D - periodic. Let 1 * u e A, U = uDG,n = {P \ P e Spec(D) and UP * U). Theorem 1.15 shows the infiniteness of the set n. Let Pen:. Since UIUP is finitely generated DG - module then it includes a proper maximal DG submodule MplUP. Since AIMp is DG - hypercentral, UIMp C\ CDG(AIMP) * < 0 > by Lemma 12.6. It follows that Uco(DG) < Mp for each x e G, because UIMp is a simple DG - module. Since it is valid for every Pen, Um(DG) < f\Ps„Mp. If we suppose that {\P&CMP = < 0 >, then Ua>(DG) = < 0 >, that is U < £DG(A), that is impossible. Hence V = f]Pex MP *< 0 >. Then UIV = (uD + V)IV. Since AlV'xs, D - periodic, AnnD(u + F ) = / * < 0 > . We have / = Pi*1... P,kl (see, for example, [NW, Chapter 1, Theorem 1.4]). This means, that Y\D{UIV) = {Pi,... ,P,}. On the other hand, by the election of V we obtain that TVD(U/V) = n is infinite. This contradiction proves the equality PDG(A) = <0>. 12.16. Lemma Let R be an integral domain, G a group, A an RG-hypercentral module Suppose B is a non-zero RG-submodule of A which is R-torsion-free.

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Then B/(B n £RG(A)) Proof

is R-torsion-free.

Put C = CRG{A). By Lemma 12.6, B f~l C * < 0 >. Let T/(B D C) be the «

- periodic part of B/(B DC). If 6 e T, then tyefiflC for some 0 * >> e /?. Clearly r is an 7?(7-submodule of A. Given g e G,weputZ>i = Z>(g- l).Then b\y = &fe- l)y = by{g- 1) = 6y, so (b\ - b)y = 0. Since B is .R-torsion-free, b\ = b. Hence b e Bf)C, and so

r=finc. 12.17. Corollary

Le/ < 0 > = Co < Ci < ...C„ < C a+ i < ...

Cr=A

be the upper RG-central series of A. If A is R-torsion-free, then Ca is an R-pure submodule of A for every a < y. 12.18. Theorem Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a non-monolithic DG - module which is D - torsion-free. If every proper factor-module of A is DG - hypercentral, then A is DG - hypercentral. Proof Suppose the contrary. Thenv! is a just non-hypercentral module. Lemma 12.15 yields that PDG(A) =< 0 >. Let F be the field of fractions for the ring D, E = A ®D F. We can consider E as an FG- module. Let U be a non-zero FG submodule of E, B=Uf]A. Then A/B = A/(A nU) = (A + U)IU. Hence A/B is D - torsion-free. Furthermore, B *< 0 > because E is an essential extension of A. It follows that A/B is DG - hypercentral. Let B = Bo , E is a non-monolithic FG - module. So we can use Theorem 12.12. Corollary 12.4 , Lemma 12.7, Theorems 12.12 and 12.18 directly imply

Just non-Hypercentral Groups and Just non-Hypercentral Modules

127

12.19. Theorem Let D be a Dedekind domain with the infinite set Spec{D), G a hypercentral group of finite 0 - rank, A a non-simple DG - module. If every proper factor-module ofA is DG - hypercentral, then A is DG - hypercentral too. The above statements are just simple translations into the module language of the main statements of [KSU 3] 12.20. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a noetherian DG - module. If A is not DG - hypercentral, then A includes a maximal DG - submodule M such that the (simple) factor-module AIM is not DG - central. Proof Let M = {B \ B is a DG-submodule of A such that AIB is not Z)G-hypercentral}. Since < 0 > e M, M * 0. Since A is a noetherian DG module, the set M contains a maximal element M. If we assume that M is not a maximal DG - submodule, then AIM is not a simple DG- module. But in this case Theorem 12.19 yields that AIM is DG - hypercentral. This contradicts the choice of M and shows that M is a maximal DG - submodule of A. 12.21. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a noetherian DG- module. If every simple factor-module of A is DG-central, then A is a DG - nilpotent module. Now we can prove some results about just non-hypercentral groups. 12.22. Theorem [KSU 3] Let G be a non-monolithic group in which every proper-factor is a hypercentral group of finite 0-rank. IfFitt(G) * < 1 >, then G is hypercentral. Proof Put A = Fitt(G), and suppose that G is not hypercentral, i.e. G is just non-hypercentral. By Theorem 10.9 either .4 is an elementary abelian p-group, for a certain prime p, or A is a torsion-free abelian group. Furthermore, CQ(A) = A. The factor-group H = GIA is hypercentral. If A is an elementary abelian psubgroup, then A is an ¥PH - hypercentral module by Theorem 12.12. If A is an abelian torsion-free subgroup, then we can apply Theorem 12.18. 12.23. Corollary [KSU 3] Let G be a non-monolithic group with Fitt(G) * < 1 >. If every proper factor-group of G is a periodic hypercentral, then G is hypercentral. 12.24. Corollary [FdeG 2] Let G be a non-monolithic periodic soluble group. If every proper-factor is nilpotent, then G is nilpotent.

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In connection with Theorem 12.22 there appears a question about the existence of the non-monolithic soluble just non-hypercentral (respectively just non-nilpotent) groups. The following simple example provides the positive answer. 12.25. Example Let A = Q2, = {-f \ m,n e Z, w * 0 and n is odd } be the additive group of 2' - adic fractions, Po = {pk | k e N} the set of all odd primes. For every pk e Po the mapping k = p^a, a e A, is an automorphism of A. Let G be the natural semidirect product of A on X ^ N < • It is not hard to prove that if B is a non-zero G - invariant subgroup of A, then the index \A : B\ is finite, moreover \A : B\ = 2' for some ( £ N . Let U and V be non-zero G - invariant subgroups of ,4 such that V < [/and \UIV\ = 2. If u e U\V, then 2w e V. We have now u(. It follows that every proper factor-group of G is nilpotent. Finally, obviously, f]mN(2')A = < 0 >, therefore the group G is non-monolithic. The results above are concerned with the non-monolithic case. Now we are dealing with the monolithic one. 12.26. Theorem [KSU 3] Let G be a monolithic group with Fitt(G) * < 1 >. Then G is just non-hypercentral if and only if the following holds: (i) G = M X H where M = Fitt{G) is the monolith of G {in particular M is abelian), H is a hypercentral subgroup. (ii) M = CG{M) andH = NG(H). (Hi) All complements to Mare conjugate in G. (iv) The periodic part Tof£,(K) is locally cyclic. (v) If for a prime p M is an elementary abelian p-group, then T is a plsubgroup. Proof Let M be the monolith of G; then M < Fitt(G) and so M is abelian. By Corollary 12.8 Fitt(G) = M is a maximal normal abelian subgroup of G, so that M = CG(M). Since G is not hypercentral, then f (G) = < 1 >, and, in particular, £(G) fl M = < 1 >. By theorem 4.5, there exists a subgroup H of M such that G = M\ H, and, moreover, any complement to M in G is conjugate with H. Since Mis the unique minimal normal subgroup of G, H = NQ(H). Put C = £(//) and let T be the periodic part of C. Then assertions (iv) and (v) follow at once from Theorems 3.1 and 3.2. The converse is immediate. In a similar way, but making use of Corollary 1.16 and Theorem 3.1 in the

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appropriate place, we may obtain the following 12.27. Corollary [KSU 3] Let G be a monolithic group with Fitt(G) * < 1 >. If G is non-hypercentral and every proper factor-group of G is a hypercentral group of finite 0 - rank, then (i) G = M \ H, where M = Fitt(G) is the monolith of G (in particular M is abeliari), H is a hypercentral subgroup of finite 0 - rank; (ii) M = CG(M) andH = NG(H); (Hi) all complements to Mare conjugate in G; (iv) M is an elementary abelian p - subgroup for some prime p; (v) C,(H) is a locally cyclic pi - subgroup. To complete these results it is worth mentioning that simple Z//-modules over hypercentral groups has been already considered in Theorems 3.1 and 3.2. 12.28. Corollary [FdeG 2] Let G be a periodic soluble group. If G is a monolithic just non-nilpotent group, then (0 G = M\H; (ii) Mis a minimal normal subgroup ofG, M = CG(M); (Hi) M is an elementary abelian p - subgroup for some prime p; (iv) H = NG(H) is a nilpotent pi - subgroup. 12.29. Corollary [FdeG 2] Let G be a soluble non-nilpotent group. If G is a just non-Nc-group, then (i) G = M\H. (ii) M is a minimal normal subgroup ofG, M = CG{M)', (Hi) H = NQ(H) is a nilpotent subgroup of class < c; (iv) all complements to Mare conjugate in G. Indeed, by Corollary 10.2 the group G is monolithic, and we can use Theorem 12.26. And, finally, the nilpotent just non-Nc-groups are described by the following result. 12.30. Theorem [FdeG 2] A nilpotent group G is a just non-Nc -group if and only if there exists a prime p such that £(G) is a locally cyclic p-group and \yc+l(G)\=p. Proof Let G be a nilpotent just non-Nc - group and put C = £(G). By Lemma 10.3, n(G) = {p} for some prime p. We claim that C is periodic. Otherwise, let z e C be an element of infinite order. For each n e N z" * 1, and so the factor-group CI < z" > is an Nc - group. Since p| n € N < z" > = < 1 > by Remak's theorem, we have G <

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< z" >, and so we obtain that G e Nc, a contradiction which proves our claim. Therefore C is a j9-group. Lemma 10.3 also yields that C is locally cyclic. Let E = Qi(C); then G/E e Nc. It follows that yc+i(G) = E, and, in particular, |yc+i(G)| =p. Conversely, it is clear that yc+\{G) is the monolith of G and it follows that G is a just non-TVc-group. TI„<=NG/

Chapter 13 Groups with Many Nilpotent Factor-Groups

In Chapter 12 we have proved the nilpotency of some groups (for example, soluble groups of finite 0 - rank and, in particular, periodic soluble groups), all proper factor-groups of which are nilpotent. In this connection the following problem comes on naturally. For what another family M of proper factor-groups of a group G the nilpotency of each factor-group from the family M needs the nilpotency of whole group Gl The following result of K. Hirsch (see, for example, [SD 2, l.C, Theorem 2]) gives us a major precedent of the consideration of this problem : If every finite factor-group of a polycyclic group G is nilpotent, then G is nilpotent too. In the paper [RD 8] D. J. S. Robinson obtained a significant amplification of this result, namely, he proved its validity for finitely generated hyper (abelian-by-finite) groups. The proof of this fundamental Robinson's theorem we will expose in this chapter. The important part of Robinson's proof is the consideration of finitely generated modules with every proper nilpotent factor-modules over a group ring IJH, where H is a finitely generated nilpotent group. This part of the proof is already in readiness. Moreover, some general situation has been considered in Chapter 12, from which the result about these modules implies as a corollary. We consider also some module version of Robinson's theorem. The following statement is an immediate corollary of the Theorem 12.19. 13.1. Theorem Let D be a Dedekind domain with the infinite set Spec(D), G a finitely generated nilpotent group, A a non-simple DG - module. If every proper factor-module of A is a DG - nilpotent, then A is DG - nilpotent too.

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132

13.2. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a nilpotentfinitelygenerated group, A afinitelygenerated DG - module. If A is not DG - nilpotent, then A includes a maximal DG - submodule M such that the (simple) factor-module AIM is not DG - central. Since the group ring DG (and, therefore, a finitely generated DG - module A) is noetherian (see, for example, [PD 1, Theorem 10.2.7.]), this result follows from Corollary 12.20. 13.3. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a nilpotent finitely generated group, A a finitely generated DG - module. If every simple factor-module ofA is DG - central, then A is a DG - nilpotent module. We consider now some module versions of Robinson's theorem. 13.4. Lemma [BC 2] Let R be a commutative noetherian ring, G a finitely generated group, A a finitely generated RG - module, B an RG- submodule of A such that AIB isfinitelygenerated asR- module. Then B isfinitelygenerated as RG - submodule. Proof. Let {gi,... ,gn} be an inverse - closed generating set of G and let A = a\RG +... + amRG. There are elements c\,... ,ct e A such that A/B= (ciR+... + c,R + B)IB. Put C = c{R+ ... + c,R; then A = C + B. For a e A v/e may write a* for an element of B such that a - a* e C. Let B\ be the RG - submodule of B generated by {(a,)* ,(cjgk)*

\l
Since R is a noetherian ring, B C\ C is a finitely generated R - submodule. In particular, B\ is a finitely generated RG - submodule. Since Cjgk e B\ + C, 1 <j 1 such that At~\ * < 0 >, but A, = < 0 >. Consider the finite series A =A0 >A\ > ... >A,-i

>A,=<0>.

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It follows that G = Ca{AilAi+i), for every 0
< 1 >= Go < Gi < ... < Gm = G such that G,/G,_i = < xtGt-\ > is infinite cyclic (see, for example, [RD 19, Theorem 5.2.20]). In particular, x\ e £(G), and so A{x\ -1)' is an TfG-submodule, for any i e N. We construct the series A = Bo >B\ > ... > Bs =< 0 >, where Bt = A(x\-\)' and s < t. Then the mapping a —• a(x\ - 1) is an RG -endomorphism, and it follows that each factor Bj/B,-\ is a finitely generated /?G-module. From the choice of Bt we deduce that fi,7B,-i can be viewed as an R(GI < xi >)-module. Simple induction on ro(G) allows us to deduce that A is a finitely generated /{-module. A Dedekind domain D is said to be a Dedekind Z - domain, if Specif)) is infinite andfor every P e Spec(D) the field DIP is locally finite. 13.6. Theorem Let D be a Dedekind Z-domain, G a finitely generated nilpotent group, and A a finitely generated DG-module. Suppose that A satisfies the following property: every DG -factor-module ofA, which is finitely generated and periodic asaD- module, is DG-nilpotent. Then A is DG-nilpotent too. Proof

Suppose that A is not DG - nilpotent. Let M = {U\ U is a DG -submodule such that A/U is not DG - nilpotent}.

We claim that M is inductive. Let L be a linearly ordered subset of M and let L be the set-theoretical union of the members of L; clearly L is a DG-submodule of A. If AIL is .DG-nilpotent, by Lemma 13.5, AIL is a finitely generated £>-module. By Lemma 13.4, there are elements a\,...,am such that L = a\DG + ... + amDG. Since £ is linearly ordered by inclusion, there is U e £ such that a\,...,am e U. Thus U = L, a contradiction, which proves our claim. Now we can apply Zorn's lemma to M. and obtain that it has a maximal element B. In other words, (i) A i = AIB is not DG-nilpotent, and (ii) every proper DG-factor-module of A \ is .DG-nilpotent. By Theorem 13.1, A\ is a simple DG-module, so Corollary 1.16 yields that Anno(A\) = P e Spec(D). Thus, A\ is & simple FG-module where F = DIP is a locally finite field. Put Q = GICG{AX). Then, by Corollary 1.21,

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C,{Q) is periodic and so £(g) is finite. Further, Q is finite too (see, for example, [RD 9, Theorem 2.24]). Therefore dimF(A\) is finite. Hence A\ is a periodic finitely generated D-module. However every such factor-module is DG nilpotent. This contradiction proves the theorem. 13.7. Corollary Let D be a Dedekind Z\-domain, G a finitely generated nilpotent group and A a finitely generated DG-module. If every finite factor-module ofA is DG-nilpotent, then A is DG-nilpotent too. Proof Any proper factor-ring of D is a finite and hence any periodic finitely generated Z)-module is finite. It suffices to apply Theorem 13.6. 13.8. Theorem [RD 8] Suppose that G is a finitely generated group having an ascending series of normal subgroups whose factors are either abelian groups or finite groups. If everyfinitefactor-group ofG is nilpotent, then G is nilpotent too. Proof

Suppose that G is not nilpotent and let M = {H | H is a normal subgroup of G such that GIH is not nilpotent}.

If C is a linearly ordered subset of M, let L be the set-theoretical union of the members of C. Suppose that GIL is nilpotent. Then, by [HP 1, pp. 421 and 426], there are a\,... ,ar e L such that L = < a\ >G ... < ar >G , and, since C is linearly ordered by inclusion, there is H e £ such that a\,... ,ar e H. Consequently, L = H, and so GIL is not nilpotent. Thus M. is inductive, and by Zorn's lemma we may choose a maximal member M of M. Put G\ = GIM. If M is a maximal normal subgroup of G; then either G\ is abelian or G\ is finite. In the latter, we note that G\ has to be nilpotent. In any case, this contradicts that Me M, so that G\ has proper factor-groups and these factor-groups are nilpotent. The group G\ has a normal subgroup P such that either P is finite or P is abelian. Suppose that P is finite. Since G\IP is finitely generated nilpotent, G\ is polycyclic-by-finite. Then G\ includes a normal torsion-free subgroup F such that G\IF is finite (see, for example, [SD 2, Section l.E]). Since G\IF is also nilpotent and we can embed G\ into (G\/P) x {G\/F), we arrive to a contradiction. Hence P is abelian. Now we can regard P as a Z(G\IP) - module. In this case Corollary 13.7 implies that P is Z(Gi//>) - nilpotent. Therefore Gi has to be nilpotent. 13.9. Corollary [RD 8] Suppose that G is a finitely generated group having an ascending series of normal subgroups whose factors are either abelian groups or finite groups. Then G is nilpotent if and only if every maximal subgroup ofG is normal in G.

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Proof Certainly, it is well-known that if G is nilpotent, then every maximal subgroup of G is normal in G (see, for example, [RD 19, Theorem 12.1.5]. Conversely, assume that every maximal subgroup of G is normal in G. Thus, if B is a normal subgroup of finite index, then GIB is nilpotent (see, for example, [RD 19, Theorem 5.2.4]). In this case, it suffices to apply Theoreml3.8 to obtain that G is a nilpotent group. 13.10. Corollary [RD 8] Suppose that G is a finitely generated group having an ascending series of normal subgroups whose factors are either abelian groups orfinitegroups. IfGIFratt{G) is nilpotent, then G is nilpotent. Let G be a group. A subgroup U is said to be pronormal in G if the subgroups U and Ug are conjugate in < U,US > for each g e G. With the help of this concept it is possible to give the following criterion of a nilpotency of a finite group: A finite group G is nilpotent if and only if every its pronormal subgroup is normal in G. In fact, clearly, every maximal subgroup is pronormal. 13.11. Corollary Suppose that G is a finitely generated group having an ascending series of normal subgroups whose factors are either abelian groups or finite groups. Then G is nilpotent if and only if every pronormal subgroup ofG is normal in G. Let G be a group. A subgroup U is said to be abnormal in G ifg e < U,Ug > for each g e G. With the help of this concept it is possible to give the following criterion of a nilpotency of a finite group: A finite group G is nilpotent if and only if it includes no proper abnormal subgroups. Indeed, if G is nilpotent, then every proper subgroup of G does not coincide with its normalizers, but every abnormal subgroup is self-normalizing (see, for example, [RD19, p. 265]). Conversely, since every non-normal maximal subgroup of any group is abnormal, it follows that every maximal subgroup of G is normal and, consequently, G is nilpotent. 13.12. Corollary Suppose that G is a finitely generated group having an ascending series of normal subgroups whose factors are either abelian groups or

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finite groups. Then G is nilpotent if and only if it includes no proper abnormal subgroup. We advert now to another similar Robinson's theorem. In Chapter 7 we met already minimax abelian groups. A soluble group G is called minimax, if it has a finite series of subnormal subgroups, every factor ofwhich is abelian minimax group. Recollecting the definition of the abelian minimax group, we can tell, that a soluble minimax group has a finite series of subnormal subgroups, every factor of which satisfies the maximal condition or the minimal condition on subgroups. It is easy to see that a soluble minimax group has a series of normal subgroups with abelian minimax factors. 13.13. Theorem [RD 4] Let G be a residually finite soluble minimax group. If everyfinitefactor-group ofG is nilpotent, then G is nilpotent too. Proof

Let < 1 > = A0
be a finite series of normal subgroups with abelian minimax factors. Let TZ = {H | H is a normal subgroup having finite index in G}. Put B = f)HeKA\H; then B is normal in G, GIB is imbedded in YlHenG/AiH>in particular, GIB is residually finite. Furthermore, if b,c e B, then b,c e A\H for every H e TZ. Since A\HIH s Ai/(A\C\H) is abelian, [b,c] eH for every H e 71. Since f]TZ = < 1 >, it follows that [b,c] = 1. Hence B is a normal abelian subgroup such that GIB is residually finite. Now using induction on n, we may assume that GIB is nilpotent. If B is finite, there is a normal subgroup U of finite index such that UC\B = < 1 >. By Remak's theorem we obtain the embedding G < GIU x GIB. Both the factor-groups G/U and GIB are nilpotent, and thus G is nilpotent too. Suppose now that B is infinite. Since a periodic abelian minimax group is Chernikov, B must be non-periodic. Let C be a free abelian subgroup of B such that ro(C) = ro(B), in particular, B/C is periodic. If follows that the set n = U(B/C) is finite. Let p i it. Then CIC is a Sylow p subgroup of BICP, so BICP = S/CP X C/CP, where S/CP is a Sylow pi- subgroup of BICP. It follows that (B/CP)P = S/CP, and therefore BPf]C= CP. Since p | tji Cp = < 1 >, the subgroup D = f] tjcBP must be periodic, and so finite. It is easy to see, that GIBP is residually finite. Using the previous arguments, we can

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obtain that GIBP is nilpotent. Let r = r0(B). For every p t n then \B/BP\ < pr. It follows that the hypercenter of GIBP with the number r includes BIBp, that is [A,rG] = [A,G,... ,G] < BP. Since it is valid for every pin, [A,rG\ < D. In r

other words, the factor-group GID is nilpotent. The subgroup D is finite, and we have already considered this case. We want to propose the following generalization of soluble minimax groups. Let G be a group, A a normal subgroup of G. We say that A satisfies the condition Max - G (respectively Min - G) if A satisfies the maximal (respectively minimal) condition for G - invariant subgroups. In other words, if A is abelian, then ZG - module A is noetherian (respectively artinian). A group G is said to be a generalized minimax, if it has a finite series of normal subgroups < 1 > = Ho < H\ < ... < H„ = G, every factor of which is abelian and satisfies Max - G or Min - G. Every soluble minimax group is obviously generalized minimax. However, the class of generalized minimax groups is much wider than the class of soluble minimax groups. The following two simple examples support this statement. Let G=wr, where a is an element of prime order p, g is an element of infinite order. Then G = A X< g > where A is an infinite elementary abelian p - subgroup. The Fp < g > - module A is cyclic, and since the ring ¥p < g > is noetherian, A satisfies Max -< g >. The second example is the following. Again consider a group ring J =FP < g >. Clearly P = (g - l)J is a maximal ideal of J. Let A be a Priifer P module. Then A = ® „ e N < a„ >, and a\g = a\, a„+\g = a„+\ +a„,n e N. Let G be a naturally semidirect product of A and < g >. Since A is an artinian J module, A satisfies Min -< g >, so that G is a generalized minimax group. 13.14. Proposition [KOS] Let G be a generalized minimax groups. If G is hypercentral then the periodic part T of G satisfies Min - G and GIT is a nilpotent minimax torsion-free group. Proof

Let < 1 > = Ho < Hi <...
be a series of normal subgroups of G, every factor of which is abelian and satisfies Max - G or Min - G. Use induction on n. If n = 1 all is trivial. Let n > 1

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and suppose that the periodic part PIH\ of GIH\ satisfies Min - G and GIP is a torsion-free nilpotent minimax group. If H\ satisfies Min - G, then T = P. Clearly T satisfies Min - G. Assume now that H\ satisfies Max - G. Since G is hypercentral, Hi has an ascending series of G - invariant subgroups, every factor of which is G - central. This series is finite because Hi satisfies Max - G. Moreover, every factor of this series is finitely generated, so that Hi is finitely generated and nilpotent. In particular, the periodic part Q of Hi is finite. A factor-group PIQ has finite 0 - rank. Let TIQ be the periodic part of PIQ. Then PIT is nilpotent torsion-free group of finite special rank (see, for example, [RD 10, Theorem 6.36]), and TIQ satisfies Min - G, so that T satisfies Min - G too. Note that a nilpotent generalized minimax group is minimax. Hence GIT is minimax. The following theorem selects a family of factor-groups in a generalized minimax group such that the hypercentrality of all factor-groups from which assures the hypercentrality of whole group. 13.15. Theorem [KOS] Let G be a generalized minimax group. If every monolithic factor-group of G is hypercentral {respectively nilpotent), then G is hypercentral (respectively nilpotent). Proof

Let < 1 > = Ho < Hi < . . . < / / „ = G

be a series of normal subgroups of G, every factor of which satisfies Max - G or Min - G. We will use induction on n. If n = 1 all is trivial. Let n > 1 and suppose that G/Hi is hypercentral (respectively nilpotent). Consider the first case when Hi = A satisfies Min - G. Let S be the G - socle of A, that is the subgroup, generated by all minimal G - invariant subgroup of A. It is easy to see that S = Mi x ... x M( where M, is a minimal G - invariant subgroup of G, 1 < / < t. Let Bj be a normal subgroup of G which is maximal with the properties XJHMJ < Bj,Bj fl M, = < 1 >, 1 < i < t. Then GIBt is a monolithic group, thus it is hypercentral (respectively nilpotent). Since B = P| 1 < / < ( 5, > ^\x<j, we have Bf]A =< 1 >. By Remak's theorem G < (Xj&GIBj) x GIA, so that G is hypercentral (respectively nilpotent). Assume now that A satisfies Max - G. We can consider A as a Z g - module where Q = GIA. Suppose that ^4 is not ZQ - hypercentral. By Proposition 13.14 Q is a hypercentral group of finite 0 - rank. Corollary 12.20 shows that A includes a maximal ZQ- submodule V such that AIV is not a G - central chief factor. Choose a maximal normal subgroup E of G such that Ed A = V. Since AIV is a G - chief factor, G/E is a monolithic group. Hence it is hypercentral (respectively nilpotent). It follows that its G - chief factor AE/E is G - central, that is

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[G,A] < E. On the other hand, [G,A] < A, thus [G,A] =

H0
< ...
be a series of normal subgroups of G, every factor of which is abelian and satisfies Max - G or Min - G. We will use induction on k. If k = 1 all is trivial. Let k > 1 and suppose that GlHi is hypercentral group. If Hi satisfies Min - G, then it has an ascending series of G - invariant subgroups < 1 > = VQ < Vi < ... Va < Va+i < ... Vy = Hi every factor of which is G - chief. Since G is Z - group, Va+ilVa is a G - central factor for every a < y. It follows that Hi lies in the upper hypercenter of G. If Hi satisfies Max - G, then Corolllary 12.21 yields that some hypercenter of G with a finite number includes Hi. Thus in this case G is hypercentral. • 13.17. Corollary [KOS] Let G be a generalized minimax group. If G is a Baer-nilpotent group then G is hypercentral. Proof

We will prove that G is Z - group. Let < 1 > = Ho < Hi < ...
be a series of normal subgroups of G, every factor of which is abelian and satisfies Max - G or Min - G. We will use induction on k. If k = 1 all is trivial. Let k > 1 and suppose that GlHi is a Z - group. Theorem 13.16 yields that GlHi is hypercentral. Let U, V be the G - invariant subgroups of Hi such that U < V and U/V is a G - chief factor. Without loss of generality we can assume that V = < 1 >, i.e. U is a minimal G - invariant subgroup of Hi. By Corollary 1.19 U is an elementary abelianp - subgroup for some prime p By Corollary 2.4 £(G/CG(U)) is a periodic pi- group since Proposition 13.14 implies that GlHi has finite 0 -

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rank. Choose an element XCG(U) e £(G/CG(U)), X i CG{U), and consider the subgroup < x, U >. If x has finite order then x = x\X2, where [xi,;t2] = 1» x\ is a p - element, xi is a p1- element. In particular, XCG{U) = X2CQ{U). Let u e U, F =< X2, U >. Then F is a finite subgroup of G. Therefore F is nilpotent. But in this case X2 e CG{U), which contradicts the choice of X2- Let now x has an infinite order. Then x' e CG(U) for some pi- number t. Put £ = < U,x >; then x' e £(£) and El < x' > is already periodic and using the previous arguments we prove that x e CG(U)- Hence in every case we obtain a contradiction, which shows that G = CG(U/V). This means that G is a Z - group. The following corollaries immediately follow from the relationship between distinct classes of generalized nilpotent groups (see, for example, [RD 10, 6.1]). 13.18.Corollary Let G be a generalized minimax group. If G is an Engel group, then G is hypercentral. 13.19.Corollary Let G be a generalized minimax group. IfGisaNthen G is hypercentral.

group,

13.20.Corollary Let G be a generalized minimax group. If G is a locally nilpotent, then G is hypercentral. Note also that for generalized minimax groups there remain valid analogies of Corollaries 13.11 and 13.12. 13.21. Lemma [KOS] Let G be a group having a normal abelian subgroup A satisfying Min - G and such that G/A is hypercentral. If every pronormal subgroup ofG is normal, then G is hypercentral. Proof By Theorem 1' of [ZD 2] A = A\ xAi where both the subgroups A \ and Aj are G - invariant; every G - chief factor of A \ is central in G (that is the upper hypercenter of G includes ^ i ) ; every G - chief factor of A-i is not central in G. Suppose that G is not hypercentral, then Ai * < 1 >. Put B = A-i. Clearly GIB is hypercentral. By Robinson's Theorem 4.5 G = B X H and every complement to B in group G is conjugate with H. Let g e G,Q = < H,H« >,C = B n Q. Since G = BH= B{HS), Q = C\H=C\W. Since G = BQ, C is a G - invariant subgroup of B. By the same reason every Q - invariant subgroup of C is also G invariant. If we assume that C has a Q - chief factor UIV with the properties CQ(U/V) = Q, then the both subgroups U, V are G - invariant, U/V is a G - chief factor and CG(U/V) = G. However, C < B, and we obtain a contradiction. Thus every Q - chief factor of C is not central in Q. Using again Theorem 4.5 we obtain that the subgroup H and H8 are conjugate in Q. This means that the subgroup H is pronormal in G. Hence H is normal and then G = B xH, a contradiction. This

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contradiction shows that B = < 1 >, thus G is hypercentral. 13.22. Theorem [KOS] Let G be a generalized minimax group. If every pronormal subgroup ofG is normal, then G is hypercentral. Proof

Let < 1 > = Ho < Hi < . . . < / / * = G

be a series of normal subgroups of G, every factor of which is abelian and satisfies Max -GOT Min - G. We will use induction on k. If k = 1 all is trivial. Let k > 1 and suppose that GIH\ is a hypercentral group. If H\ satisfies Min - G, then we can apply Lemma 13.21. Let now Hi satisfies Max - G. By Proposition 13.14 GIH\ has finite 0 - rank. Let B be a maximal G - invariant subgroup of Hi; then GIB satisfies the conditions of Lemma 13.21. By this lemma GIB is hypercentral, i.e. H\IB is a G - central factor. By Corollary 12.21 some hypercenter of G with the finite number includes H\. Thus in this case G is hypercentral. 13.23. Corollary [KOS] Let G be a generalized minimax group. IfG includes no proper abnormal subgroups, then G is hypercentral.

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Chapter 14 Groups with Proper Periodic Factor-Groups

Chronologically, just infinite groups (that is the just non-finite groups) were one of the first investigated classes of just non-A"-groups. They are natural generalization of infinite cyclic groups. In fact, every non-identity subgroup of an infinite cyclic group has finite index. Moreover, the converse statement is also correct: if every non-identity subgroup of an infinite group has finite index, then this group is cyclic (Yu. G. Fedorov, [FYu]). The theory of just infinite groups has its own specificity. If the majority of results on just non-A"-groups for other classes X concern with generalized soluble groups, the development of the theory of just infinite groups requires to work with not only generalized soluble groups. As an illustration to that we can point the examples of just infinite finitely generated p-groups constructed by R. I. Grigorchuk [GRI 1, GRI 2] and N. Gupta, S. Sidki [GUS], the study of distinct properties of which will be carried out intensively now. Let G be a just infinite group, L = HP(G) the Hirsch-Plotkin radical of G (that is the product of all locally nilpotent normal subgroups of G). Obviously Fitt(G) < L, so, if L * < 1 >, then GIL is finite. By [WJ 1] L satisfies the maximal condition for normal subgroups (the condition Max - n). In turn, a locally nilpotent group satisfying Max - n is finitely generated [GV]. Similarly to other results of this book, we may obtain 14.1. Theorem [McC 1,WJ 2] Let G be a group with HP{G) * < 1 >. Then G is a just infinite group if and only ifG satisfies the following conditions: (/) A = HP(G) is a maximal normal abelian subgroup ofG {in particular, A = CG{A)); (»') A is torsion-free and finitely generated; (Hi) GIA is finite ; (/v) A is a just infinite ZG-module.

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144

D. McCarthy in papers [McC 1, McC 2] obtained some details of the structure of just infinite groups with a non-identity locally nilpotent radical; in particular, he described metabelian just infinite groups and obtained some properties of automorphism groups of just infinite groups. On the other hand, the structure of just infinite groups with an identity locally nilpotent radical is rather complicated. The following result points out on this statement. 14.2. Theorem [McL] There exists properties: (z) G is periodic and locally soluble (ii) G is residually finite ; (Hi) ifH is a normal subgroup ofG, (Here G w stands for the (/ + l)th term

a just infinite group G with the following ; then H = G^ for some i e N. of the descending derived series of G).

Proof For each i e N we construct inductively a finite soluble group G, with a unique minimal normal subgroup Mt. We start by taking G\ to be the symmetric group of degree 3 and M\ the unique Sylow 3-subgroup of Gi. Suppose we have constructed G; and take p to be an arbitrary prime not dividing the order of G,: p <£ I1(G,). Let Ri = FpGt, then Rit being considered as a right Rt - module, is semisimple by Mashke's theorem (see, for example, [CUR 1, Theorem 10.8]). If each simple submodule of/?, is non-faithful, then M, lies in the kernel of each of the corresponding irreducible representations of G,; therefore Rt is not faithful. This is impossible; so there must exists a faithful simple submodule M,+i of /?,-. Looking at G, as an automorphism group of the additive group Mi+\ we form the corresponding natural semidirect product Gi+\ of M,+i with G,. Note that G,+i is a finite soluble group, which includes G, properly and in which Ml+\ is its unique minimal normal subgroup since it is self-centralized. Let G be the union of the chain Gi < G 2 < ... < G, < .... Then G is a periodic locally soluble group. Let H be a proper non-identity normal subgroup of G. If H n G,• * < 1 >, then M, < H because M, is the monolith of Gt, and if i > 1 Hr\G,=Hn

(M,G,-,) = M,{Hf)

GM).

Lety be the least number such that / / f l Gy * < 1 >. Evidently H is generated by the sequence of subgroups H n G,, H C\ GJ+\,... ,and so by H n G7, Mj+i,MJ+2,.... Clearly H is generated by the sequence of subgroups H fl Gj,Hf) Gj+\,..., and so by HHGj, Mj+u Mj+2, ... . If j > 1, then / / n G 7 = M y because HC\Gj = Mj(HnGj-i), and if j = 1, then either HC\G\ = M\ or H > Gu

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145

which clearly implies that H = G. It follows that H = < Mj,Mj+\,... >, which is exactly the (/+ l)th term of the derived series of G. Thus, these all are proper non-identity normal subgroups of G and then G is clearly just infinite. However, G is insoluble because any G® never vanishes. Further G = G^G, for every i e N, and thus |~) /eN G w = < 1 >• In particular, G is residually finite. Extending this case, B. Hartley [HB 1] constructed an uncountable periodic locally soluble just infinite group whose Sylow subgroups are finite. In the torsion-free case the situation is also complicated . The following example developed by P. Hall supports this statement. Let/? be an odd prime and Z{px) be the ring of all/7-adic integers. Choose the numbers h, k, t e Z such that (1)

n> 0,k> 0,t> 0,h + k> t.

We denote by H = H(h,k,t) the set of all matrices

with coefficients in Zip™) and with the determinant (2)

ad- be = 1

satisfying (3)

b = 0{modph),c = 0(modpk),d = \{modp').

Then a = (1 + bc)ld = \{modp') by (2), so that

i

f

*-'-{-c

d

-b\

a J*"

for every % e H. Hence H is a group. It is convenient to write G = //(0,1,1) and Gm = H{m,m,m). Since the natural homomorphism of Z(p°°) onto Z(pco)//7mZ(pco) induces a homomorphism of G whose kernel is Gm , it turns out that Gm is a normal subgroup of G, and besides \G : Gm\ = P3m~2- In general, if H = H(h,k,t), we have H < G and it is easy to see that \G:H\= ph+k+'~2. Actually Gm < H when m > max(h,k,t). Put K = < %H >, where 1 <£ x e H. Also, given x,y,u e Z^""), we will write

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146

(4)

P{x) -

Then it is rather straightforward to deduce the expressions (5)

l+xcd + x2+c2

[/?(*),*]

x{d2-\-Yxcd) 1 —xcd

Taking the contragradient matrices, we obtain (6)

[y(y),x] = (

X yab

~yhl

~

\^ y{a2 - 1 +yab) 1 +yab +y2b2

\ J

With these ingredients, we have 14.3. Lemma [HP 4] There exists some p-adic number 0 * XQ such that P(x0) e K. Proof Suppose that m is a number such that Gm < H. If c = 0 and d = 1, then X = P(b) and b * 0 since # * 1. If c = 0 and d * 1, then AT contains the element [jS(Pm),l] = P(pm{

Xi £ ^- Using (5) we have 1 - d\ = pmcd{2 -pmcd) so that v(l - d\) = m + k\ because/? is odd and m > 0. Similarly pmc\d\ = -p2mc2{\ -pmcd), which give us v{pmc\d\) = 2(m + k\). But 62 = Pm(d\ - 1 +pmc\d\), and so v(bj) = 2m + k\, showing incidentally that b2 * 0. Finally, v(l - a\) = v(l - a2) = v(pmadi) = 2{m + ki) and v(«2) * 0. Hence if y = (1 - a\)l{aibi), then v(y) = &i = v(c) and so y{y) e //. It follows that 13 = lr(y),X2] =

a\

-62(1 - a-^lai

0

a?

belongs to K. Since a\ =t= 1, X3 * 1- Thus we have the case c = 0 already treated. This completes the proof. 14.4. Theorem [HP 4]

IfK is an arbitrary non-identity normal subgroup ofH,

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147

then HIK is a finite p-group. Proof Define the subgroups B„,C„,D„ as follows: B„ consists of all B(x) with v(x) > n; C„ consists of all y(y) with v(y) > «; and D„ consists of all 8{u) with V(M - 1) > n. If uv - xy = 1 we have (7)

= B(xly)y(yu)8(u).

Hence HhX, = BhCkD,. By Lemma 14.3 K contains a matrix j8(*o) for some xo * 0. Hence if v(u~l)>t, then K also contains [B(xo),S(u)] = B{XQ{UI - 1)). But every x e Z(px) which satisfies v(x) > t can be expressed in the form u2 - 1 for some such u , because p is odd. Hence B„ < K for n = v(xo) + t. Next using (5) and (7) we can obtain [0to,rOO] =

B(x2yz-')y(-xy2z)5{z)

where z = 1 - xy. Taking x = p" and y = pk, we have y(y) e H and /J(x) e K, because B„ < K. Hence [B(x),y(y)] e A". But B{x2yz~x) e / : since B„ < K and z = I -xy. Hence (8)

X4

= 7(-*y 2 z)<5(z)

z"1

0

2

-x y z

and ^ 5 = [y(y),X*] = r(y( z 2 - 0 ) He in^. Here z = 1 -p" + k , so that ^ 5 * 1 - Hence K contains y(yo) for some yo * 0. Using the formula [y(yo),<5(w)] = r(yo(w-2 - 1)), we obtain that C„, < K where «i = v(yo) + 1, just as for B„ < K. Finally, B„ < K, (8), z = I -xy, and inclusion C„, < K yields that D„2 < K where «2 = « + »i. Hence K includes a subgroup Gr = BrCrDr for a large enough r. Since G/Gr is a finite p-group and Gr < K < H < G, it follows that HIK is also afinitep-group. Since HmeN Gm = < 1 >, / / is residually finite. The subgroup F, generated by the matrices and is a subgroup of H for all large m. But F is a free group of free rank 2 [BJ].

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Characterizations of just infinite groups with identity locally nilpotent radical were also obtained by J. S. Wilson [WJ 2]. These results express important properties of just infinite groups, in particular, properties of the Grigorchuk groups and the Gupta-Sidki group. However, as it seems, they have remained unnoticed for experts who study these examples. Since in our book we only concern with groups G satisfying the property Fitt{G) * < 1 >, and Wilson's results require rather special techniques, we simply quote the most fundamental results of the mentioned paper omitting their proofs. 14.5. Theorem [WJ 2] Let G be a just infinite group with HP(G) = < 1 >. IfH is a subnormal subgroup of G, then there exists a subnormal subgroup C such that < H,C > = Hx C and the index \G : HC\ is finite. 14.6. Corollary [WJ 2] Let G be a just infinite group with HP{G) =< 1 >. If H and K are two subnormal subgroups ofG, then < H,K > is subnormal too. In particular, the set Ssn(G) of all subnormal subgroups ofG is a sublattice of the lattice of all subgroups ofG. IfH,K

e SS„(G), then we write H~ K if\H : Hf\K\\K

: Hf)K\ is finite.

Obviously ~ is an equivalence relation. Moreover, ~ is a congruence on Ss„(G).

The factor - lattice C(G) ofSs„(G) lattice ofG.

by the congruence ~ is called the structure

14.7. Theorem [WJ 2] Let G be a just infinite group with HP{G) = < 1 >. If the lattice C(G) is infinite, then it is isomorphic with the lattice of the closed-and-open subset of the Cantor's ternary set. If C(G) is finite, then G satisfies the maximal condition for subnormal subgroups. The most near to finite groups are polycyclic-by-finite and Chernikov groups. Just non- (polycyclic-by-finite) groups will be considered in the following chapter. Just non-Chemikov groups we shall consider now. Moreover, some general situation will be considered here. A group G is called hyperfinite, if G possesses an ascending series of normal subgroups with finite factors. Clearly, every Chernikov group is hyperfinite, every periodic soluble-by-finite group of finite abelian section rank (in particular, of finite special rank) is hyperfinite. Note also that hyperfinite groups are exactly the periodic FC hypercentral groups. Also it is obvious, that the class of all hyperfinite groups is a

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149

formation. Now we will consider just non-hyperfinite groups. The study of such groups needs to be split on the following two cases: the FC center is non-identity and the FC - center is identity. 14.8. Theorem [KSU 4] Let G be a group with FC(G) *< 1 >. Then G is a just non-hyperfinite group if and only if G includes a normal abelian subgroup A satisfying the following conditions: (0 A is a torsion-free subgroup of finite 0 - rank; (ii)A = CG(A); (iii) A isX - irreducible in G; (iv) GIA is finite. Proof Let 1 * b e FC(G),B = < b >G ; then \G : CG(B)\ is finite and B is central-by-finite. By Schur's theorem (see, for example, [RD 9, Theorem 4.12]) [B,B] is finite. Since G does not include finite normal subgroups, it follows that [B,B] = < 1 >, i.e. B is abelian. In this case the set T of all elements of B, having finite orders, is a subgroup of B. Since B is finitely generated, the subgroup T is finite. Clearly T is normal in G, and again T = < 1 >. In other words, B is a free abelian subgroup of finite rank. Put C = CG(B). Since GIB is hyperfinite, [C,C] is periodic (see, for example, [RD 9, Corollary to Theorem 4,12]). However, in this case [C,C] V\B = < 1 >, which contradicts Lemma 10.3. This means that [C,C] = < 1 >, i.e. C is abelian. By the same reason, C is torsion-free. Let A be a maximal normal abelian subgroup of G including C. Using the previous arguments, we can prove that CG(A) is abelian and torsion-free. It follows that CQ(A) = A. Since CG{B) = C . Let 1 * e e A n FC(G),E = < e >G . Since GIE is hyperfinite, AIE is periodic. This means that A has finite 0 - rank and A is Z- irreducible in G. Conversely, assume that G includes a normal abelian subgroup A satisfying the conditions (i) - (iv), H is a normal non-identity subgroup of G. Since A = CQ(A), A fl H * < 1 >. By (iii) AI(A n H) is periodic, so GI(A fl FT) is periodic and abelian-by-finite. Clearly, such groups are hyperfinite. 14.9. Corollary Let G be a group with C, (G) * < 1 >. Then G is a just non-hyperfinite group if and only ifG is a torsion-free locally cyclic group. 14.10. Corollary Let G be a group with FC(G) i= < 1 >. Then G is a just non-Chernikov group if and only if G includes a normal abelian subgroup A satisfying the following conditions: (i) A is torsion-free minimax subgroup; (ii)A = CG(A);

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{Hi) A is Z- irreducible in G; (iv) GIA is finite. These assertion are contained in Theorems B and C of paper [FdeG 1]. Now we will consider just non-hyperfinite groups with identity FC - center. 14.11. Proposition [KSU 4] Let G be a just-non-hyperfinitegroup 'with FC{G) = < 1 >. IfFitt(G) * < 1 >, then either Fitt(G) is torsion-free abelian or Fitt(G) is an elementary abelian p - subgroup for some prime p. Moreover, Ca(Fitt(G)) = Fitt{G). In fact, since FC{G) = < 1 >, G is a just non-FC-hypercentral group, and we may apply Theorem 10.5. 14.12. Lemma [KSU 4] Let G be a just non-hyperfinite group with FC{G) = < 1 >, A a non-identity normal abelian subgroup ofG,FICa{A) a finite normal subgroup of G/CG(A).

ThenCA(F)

=<

1 >.

Proof Put Q = GICG{A),HICG{A) = CQ{FICG{A)); then H is a normal subgroup of finite index. Suppose the contrary, let C = CA(F) * < 1 >. Let 1 * g e F\CG(A). Consider the mapping g * < 1 >. The factor-group H/CA(g) is hyperfinite, because it is a section of the hyperfinite group GIC. Since g £ CG(A), A * CA(g). By Corollary 3.4 AICA{g) includes a non-identity finite H - invariant subgroup BICAig)- Put YICA(g) = CH(B/CA(g)). If hsY,beB, then bh = bd for some element h d e CAig). Similarly, g = gu for some u e CQ{A). Now we obtain {b, because B * CAig). This contradiction proves that CA{F) = < 1 >. 14.13. Lemma [FdeGK 3] Let G be a hyperfinite locally soluble group. If there is a prime p such that 0PiG) = < 1 > and Opi{G) is finite, then G is finite. Proof. Assume that G is infinite. Clearly the normal subgroup C = CciOpiiG)) has finite index, so that CliC fl OpiiG)) is infinite. Since the last factor-group is a periodic hyperabelian group having no non-identity normal /^'-subgroup, it includes a non-identity abelian G-invariantp-subgroup PI{C C\ 0 P /(G)). Then P is

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151

nilpotent and hence Op{P) is a non-identity normal p-subgroup of G. This contradiction shows that G is finite. 14.14. Corollary Let G be a hyperflnite (locally soluble)-by-flnite group. If there is a prime p such that 0P(G) = < 1 > and Opi(G) is finite then G is finite. Let G be a just non-hyperfinite group, A = Fitt(G) and suppose that ,4 * < 1 >. Put M.(A) = {B\B is a non-identity G - invariant subgroup of A} The following two cases naturally appear here: M.(A) = < 1 > (the non-monolithic case); M(A) * < 1 > (the monolithic case); 14.15. Lemma [KSU 4] Let G be a just non-hyperfinite group with FC(G) = < 1 >, A = Fitt(G) * < 1 > and suppose that A is an elementary abelian p subgroup for some prime p. If G is (locally soluble)-by-finite, then G is a monolithic group. Moreover, A is the monolith ofG. Proof It is sufficient to prove that M(A) = 0. Indeed, in this case A is a minimal normal subgroup of G, and Lemma 10.3 proves that ,4 is the monolith of G. Suppose the contrary; let M(A) + 0, and let B e M(A). If we assume that Op(GIA) * < 1 >, then Corollary 3.4 yields that GIA includes a non-identity normal finite p - subgroup KIA. But in this case the subgroup K is nilpotent (see, for example, [RD 10, Lemma 6.34]), in particular, CA(K) * < 1 >. This contradicts Lemma 14.12. This contradiction shows that Op(GIA) = < 1 >. Suppose now that QIA = Opi(GIA) is infinite. Corollary 3.4 shows that AIB includes a non-identity finite G - invariant subgroup EIB. Put UIB = CGIB(EIB); then U is a normal subgroup of finite index. It follows that \Q : Q R U\ is finite, in particular, Q\IA = (Q f] U)IA is infinite. Using again Corollary 3.4 we obtain that Q\IA includes a non-identity finite G - invariant subgroup VIA. By Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]) E includes a V - invariant subgroup D such that E = B x D. By the selection of V, [D, V] < B. But [D,V] < D because D is V - invariant, that is [D, V\ < B n D = < 1 >. It proves that CA(V) * < 1 >, which contradicts Lemma 14.12. This contradiction shows that Opi(GIA) is finite. But in this case GIA is finite by Corollary 14.14, and hence FC(G) i= < 1 >. This final contradiction proves that M(A) = 0. 14.16. Lemma [KSU 4] Let G be a just non-hyperfinite group, A = Fitt(G) * < 1 > and suppose that A is torsion-free. Then FC(G) * < 1 >.

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Proof Suppose the contrary, let FC(G) = < 1 >. Proposition 14.11 yields that A is abelian and CG(A) = A. If we assume that ro(A) is finite, then G/A is isomorphic to some periodic subgroup of GLr(Q), where r = ro(A). However periodic subgroups of GLr(Q) arefinite(see, for example, [WB, Theorem 9.33]), so G/A is finite and hence FC(G) * < 1 >. This contradiction shows that ro(A) is infinite. Let K* = KJA be a finite normal subgroup of G* = G/A. FutE = A ® z . We may consider £ as a QG* - module. Moreover, E is a simple QG* - module, because A is Z - irreducible in G. Let 1 * u s A and U a QK* - submodule, generated by u. Since K* is finite, dimq U is finite, so that U includes a simple QK* - submodule V. Lemma 3.5 assures the existence of a subset S such that (in additive notations) E = ®xeS Vx. Since E is a Z - essential extension of A, EIA is periodic. It follows that and V/(V(~)A) is periodic, in particular, Vf\A * < 1 >. Let 1 * w e Vf]A,W = <w>K; then W and FC(G) = < 1 >. If G is a just non-hyperfinite group, then (i) A is an elementary abelian p - subgroup for some prime p; (ii) A is the monolith ofG. In fact, Lemma 14.16 and proposition 14.11 give that/I is an elementary abelian p - subgroup for some prime p. Now we can use Lemma 14.15. 14.18. Theorem [KSU 4] Let G be a (locally soluble)-by-finite group with FC(G) = < 1 >, andFitt(G) = A * < 1 >. Then G is a just non-hyperfinite group if and only ifG satisfies the following conditions: (i) A is an infinite elementary abelian p - subgroup for some prime p; (ii) A is the monolith ofG (i.e. A is the unique minimal normal subgroup ofG); (Hi) A = Ca(A); (iv) G = A X H, where H is a hyperfinite subgroup;

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153

(v) the complements to A in G are conjugate with H; (vi) ifS = Socab(H), then S is a pi-subgroup; (vii) S includes a subgroup Q such that SIQ is locally cyclic and Coren(Q) = < 1 >. Proof Let first G be a just non-hyperfinite group. Corollary 14.17 and Proposition 14.11 imply (i) - (iii); Robinson's Theorem 4.5 assures (iv), (v), and Theorem 3.15 proves (vi) and (vii). The converse is immediate. 14.19. Corollary Let G be a group with FC(G) = < 1 > and Fitt(G) =A± < 1 >. Then G is a just non-Chernikov group if and only if G satisfies the following conditions: (i)A is an infinite elementary abelianp - subgroup for some prime p; (ii) A is the monolith ofG (i.e. A is the unique minimal normal subgroup ofG); (iii)A = CG(A); (iv) G = A \ H where H is a Chernikov subgroup; (v) the complements to A in G are conjugate with H; (vi) ifS = Socat(H), then S is a pi-subgroup, in particular, the divisible part of H is a pi- subgroup; (vii) S includes a subgroup Q such that SIQ is locally cyclic and Coreu(Q) = < 1 >. For the case of a soluble group G the statements (i) - (iv) and (vi) have been proved in [FdeG 1]. The following question arises in connection with the results of this Chapter and Chapter 12 . Question 6 Describe the structure of a group G with Fitt(G) * < 1 >, every proper factor-group of which is a FC - hypercentral group of finite 0 - rank.

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Chapter 15 Just non-(Polycyclic-by-Finite) Groups

The classes V of all polycyclic-by-finite groups and C of all Chernikov groups are the most near extensions of the class T of all finite groups and they have been studied intensively by several authors. Since these groups are finite extensions of soluble groups with the two main chain conditions Max and Min respectively (see [RD 9, 3.1]), the classes V and C are dual to each other in some sense. Also V fl C = T. In the previous chapter we have already considered just non-Chernikov groups. This chapter will be devoted to just non-(polycyclic-by-finite) groups. J. R. J. Groves [GJ 1] initiated the study of these groups. More specifically, he considered the metanilpotent case. The description of soluble just non-polycyclic groups has been obtained by D. J. S. Robinson and J. S. Wilson [RW]. We have already remarked, that this paper [RW], saturated with new ideas, constructions, and results, has played a great role in the theory of just non-A'-groups. In Chapter 8 we have already exposed some results of [RW]. The current chapter exposes other results of [RW] devoted to just non-polycyclic groups. We start with some elementary properties of just non-(polycyclic-by-finite) groups, which we shall freely use in what follows. 15.1. Lemma [RW] (i) Every finitely generated soluble non-polycyclic group has a just non-polycyclic factor-group. (ii) Every just non-{polycyclic-by-finite) group satisfies Max-n, the maximal condition on normal subgroups, and is finitely generated. (Hi) IfG is a just non-(polycyclic-by-finite) group and Fitt(G) * < 1 >, then G is abelian-by-polycyclic-by-finite. The last statement follows from Theorem 10.10.

155

156

Just non-X-Groups

We need the following ring-theoretical lemma. 15.2. Lemma [RW] Let R be a commutative ring, A an R-module having a finite composition series, S = SOCR(A). If A/S and S have no isomorphic simple submodules, then A = S. Proof Suppose that A =t= S. Let BIS be a simple R - submodule of A/S. Suppose that S = Si © ... © S„ where St is a simple /^-submodule of A, 1 < /' < n, and put I = Annit(B/S), Ij = AnnR{Sj), 1 <j < n. By Chinese Remainder Theorem (see, for example [LS, Chapter 11, Section 2]) there is an element x e R such that x e I, x e I + Ij for each j , 1 <j. Suppose that Kerfx n S * < 0 >. Then S fl Kerfx includes a simple ^-submodule U. Since S = ©!<.-<„ Sy, there is a number t such that U = 5 r . In particular, AnnR(U) = /,. Since x = 1 + 7 where >> e AnnR(U), we can write Ufx = Ux= U(l+y) = U+Uy= U. On the other hand, t/ < #er& so Ifc = < 0 >. This contradiction shows that Kerfx fl S = < 0 >. Since 5/5 is a simple /{-submodule, B = S(B Kerfx and £er/i is a simple i?-submodule. However in this case Kerfx < SOCR(A) = 5. This contradiction shows that .4 = S. 15.3. Proposition [RW] Let G be a just non-(polycyclic-by-finite) group, A = Fitt(G), H = GIA. Then A is a just infinite ZH - module. Conversely, let H be a polycyclic-by-finite group, A a just infinite ZH - module such that CH{A) — < 1 >. If A is not Z-finitely generated then every extension G of A by H is a just non-(polycyclic-by-fmite) group such that Fitt{G) = A. Proof Let G be a just non-(polycyclic-by-finite) group. By Theorem 10.10 either A is an elementary abelian p-subgroup for some prime p, or A is torsion-free abelian. Let us begin with the periodic case, so suppose that A is an elementary abelian p-group, p is prime. Given a non-identity G-invariant subgroup B of A, we have that AIB is finite since GIB is polycyclic-by-finite. Thus the condition (JI 1) is clearly satisfied and so it suffices to check the condition (JI 2). Let Tia(A) = {B I B is a non-identity G - invariant subgroup of A}, and put C = f]Ha(4). Suppose that C * < 1 >. By the election, C is a simple WpH - submodule of A. But these modules are finite [RJ 1]. This contradiction shows that p | W o ( ^ ) = < 1 >. Therefore we have just checked (JI 2) and hence A is even a just infinite Fp/Z-module in this case. Let now A is a torsion-free and abelian. We split the proof in two

Just non-(Polycyciic-by-Finite) Groups

157

complementary cases assuming first that A is Z-irreducible. If B e HG(A), then GIB is polycyclic-by-finite. Since AIB is periodic, AIB is finite. Suppose again that C * < 1 >. Corollary 1.19 yields that C is an elementary abelian p-subgroup for some prime p, what is impossible because A is torsion-free. So P | 7 7 G 0 4 ) = < 1 > and A is a just infinite 7LH - module. Therefore we may assume that A is not Zirreducible. Then A includes a G - invariant subgroup B such that AIB is not periodic. Let E\IB be the periodic part of AIB, then E\IB is finite and AIE\ is a free abelian group of finite 0 - rank. Suppose that E\ is also not Z - irreducible. Then E\ includes a G - invariant subgroup Ei * £i such that £i/£2 is a free abelian group of finite 0 - rank. Otherwise, we may iterate the above construction and build up an infinite (proper) descending chain A > E\ > E2 > ... > E( > ... such that A/E, is a free abelian group of finite 0 - rank and ro(A/Ej) < ro(AIEi+i) for any i e N . Letp be a prime. Then \{AIEi)l{AIE,)p\ < \{AIEt+x)l{AIEi+\)p\ for every n e N. It follows that AIAP is infinite. Ap * < 1 > since A is torsion-free. Hence GIAP is polycyclic-by-finite, and therefore AIAP is finite. This contradiction shows our claim and then A includes a Z-irreducible Z//-submodule U, and All] is free abelian of finite rank. By the arguments above, U also becomes a just infinite Z//-module; in particular, U is a finitely generated Z//-module. By Corollary 1.8 there is a finite set n of primes such that U e A(Z,Tt). Suppose that q £ n. Lemma 1.6 implies the equality rz(U) = dim¥q{UIUq). Since Uq * < \ >, GIW is polycyclic-by-finite, in particular, UIUq is finite. It follows that r%(ll) = r0(U) is finite. Since All! is finitely generated, and ro(A) is finite too. In this case A has a finite series of 7LH - submodules U = Uo < Ui < ... < U„ = A such that Ui is a pure subgroup of A and a Z/f - module C/,7C/,--i is Z - irreducible for any ;', 1 < i < «. It follows that G/CG(U) and G/CG(t/,/f/,-i) are abelian-by-finite (see, for example [WB, Lemma 3.5]). Put

L = cG(£/<>) n cG(UiiUo) n... n c G (t/ n /f/„-i). Then G/i is abelian-by-finite by Remak's Theorem. Moreover, LIA is nilpotent (see, for example, [KW, Theorem l.C.l]). By construction L is nilpotent itself, so that L < Fitt{G) = A. In other words, GIA is abelian-by-finite. Let K be a normal subgroup of finite index such that H\ = KIA is abelian. We consider A as Z//j module. Put V = A z Q , Vo = t/o ®z Q. Since £/o is Z- irreducible, V0 is a simple QH - module. By Proposition 3.6 Vo includes a simple Q//i - submodule W such that VQ = Wx\ © ... © fFx, for some elements x\,...,xt e H. Let

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D = i/o fl W. Obviously D *< 1 >. Assume that D is finitely generated as subgroup. Choose the elements y\,...,yt such that x\ =y\A,... ,xt — ytA. Then Dy'... Dy' is a Z// - submodule which is a finitely generated as a subgroup. Since Uo is Z - irreducible, UolDy'...Dy' is finite. Hence f/o is finitely generated as a subgroup, which is impossible. This contradiction shows that the subgroup D is not finitely generated. Let S/V0 be a simple Q//i - submodule of V/V0, R = SC\A. Then R/U0 is a Z irreducible QHi - submodule of A/UQ. Since A/Uo is finitely generated as a subgroup, Z//i - modules DXi and 7?/t/o are not isomorphic. Therefore CHl (DXl) * CHl (R/Uo), \ and G be an extension of ^ by H. Let Xbe a non-identity normal subgroup of G. If we assume that Xf\A = < 1 >, then X = X4A4 is isomorphic with some subgroup of CH{A) = < 1 >. So XC\A * < 1 >. It follows that/l/(Xn^) is finite, therefore G/(XC\A) and also G/Xis polycyclic-by-finite. Finally, let A] = Fitt{G). Then A , H = GIA. If A is not periodic, then (i) A is a minimax abelian torsion-free subgroup; (ii) the ZH- module A is just infinite; (Hi) H is abelian-by-finite; (iv) there exists a free abelian subgroup X of finite 0 - rank such that XC\A = < 1 > and the index \G : XA\ is finite. Moreover, if G splits over A, the complements of A fall into finitely many conjugate classes. Proof By Theorem 10.10 A is a torsion-free abelian subgroup. By Proposition 15.3 A satisfies (ii), in particular, A is a finitely generated Z// - module. By Corollary 1.8 there exists a finite set of primes n such that .4 € A(Z,n). Choose a prime q £ n; then by Lemma 1.6 r%(A) = ro(A) = dimtq(AIAi). Since A is torsion-free thus Aq *< 1 >, so GIAq is polycyclic-by-finite. In particular, AIAq is finite, and therefore ro(A) is finite. Together with ,4 e A(l<,n) this gives that ,4 is a minimax subgroup. Corollary 7.8 implies that H is abelian-by-finite and finitely generated. Let K be a normal abelian torsion-free subgroup of// such that H/K is finite. We have K < FC(H). If [A,K] * A, then A/[A,K] is finite. Hence the all conditions of Theorem 5 of [RD 13] are realized. By this theorem we obtain that

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159

G satisfies (iv). 15.5. Theorem [RW] Let G be a just non-(polycyclic-by-finite) group with A = Fitt(G) * < 1 >, / / = GIA. Suppose that A is periodic andFC(H) is infinite. Then (/) A is an elementary abelian p-subgroup for some prime p; (ii) H includes a normal torsion-free abelian finitely generated subgroup L of finite index; (Hi) there is an element 1 =£ x e L such that A is¥p < x > - torsion-free and ¥p < x >-minimax; (iv) G includes a free abelian subgroup X of finite 0 - rank such that X DA = < 1 > and the index \G : XA\ is finite. Moreover, if G splits over A, the complements ofA fall intofinitelymany conjugacy classes. Proof (i) follows from Theorem 10.10. Put F = FC(H). Then F is a finitely generated FC - group, so F is central-by-finite. Moreover, the index \H : Cn(F)\ is finite. If C = Q(F), then C is finitely generated abelian subgroup and there is a number t e N such that D = C is torsion-free. Since D < FC(H), the subgroup K = CH(D) has finite index. By Proposition 15.3 A is a just infinite"LH- module. We may apply Proposition 6.9 to deduce that A includes an ¥PK - submodule B such that AIB is a just infinite FPK - module. Moreover, if T is a transversal to K in H, then A/By is a just infinite ¥PK - module for every y £ T and A < ®y&TAIBy. Suppose that£>/C/)C4/B) is finite. Then by DICD(AIBy) = D/y-lCD(A/B)y = (y-lDy)/(y~iCD(A/B)y) s D/CD(A/B), DICo(AIBy) is finite too any y € T. Proposition KICK(A) < XyeTKJCK(A/By). It follows that

6.9

implies

that

D < Xy&T(DCK(A/By)/CK(A/By)) = XyeTDICD(AIBy). In particular, D is finite because CD(A) = < 1 > by Theorem 10.10. This contradiction shows that DICD(AIB) contains elements of infinite order. Since D < C(K), proposition 6.18 and Corollary 7.19 imply that K/CK(A/B) is abelian-by-finite. From the embedding K/CK(A) < Xyt=TK/y~lCK(A/B)y we can see that KJCK(A) itself is abelian-by-finite. By Theorem 10.10 CK(A) = < 1 >, therefore K and H are abelian-by-finite. Fix a normal torsion-free abelian subgroup L of//having in //finite index, so that we have just proved (ii). Proceeding as above, we may conclude that A includes an ¥PL - submodule U such that Al'U is a just infinite FPL - module. Moreover, if S is a transversal to L in H, then A/Uy is a just infinite ¥PL - module for each y e S and A < 0 SA/Uy. In the same way LICL(AIU) is not periodic. Hence L contains an element x\ such that < xi > C\CL(AIU) =< 1 >. By Corollary 7.20 AIU is ¥P < x\ > - torsion-free

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and ¥p <x\ > - minimax. Let S = {yi,... ,yi}. We proceed by induction on the number t of summands, starting with the case t = 1. Let X2 be an element of L such that <x2 > nCL(A/Uyi) = < 1 >. If < xi > f]CL(A/Uyi) = < 1 >, then AlUyi is ¥p < x\ > - torsion-free and ¥p < x\ > - minimax. In this case AIU ® AIUy\ become also ¥p < x\ > - torsion-free and F^ < x\ > - minimax. In this case we put x = xi. If < x\ > rvCi.(A/Uyi) * < 1 > but < X2 > p[Ci{AIU) = < 1 >, then put x = xi. Again AIU © A/Uyi is Fp < x > - torsion-free and Fp < x > minimax. Thus we have reduced the proof of (iii) to the case in which < x\ > f] CiiAIUyi) * < 1 > and < x2 > P[CL(AIU) * < 1 >. Then there are numbers ki,k2 such that x\' e CL(AIUyx), and x*2 e CL(AIU). Put x = x\'xk2\ Then x acts on AIU as x{' does, and on AIUy\ as x22 does. It follows that AIU © AlUy i is ¥p < x > - torsion-free and ¥p < x > - minimax. The general case follows inductively. As in Theorem 15.5, the assertion (iv) follows from [RD 13, Theorem 5]. To complete the periodic case, note that, since G is not polycyclic-by-finite, the complementary case to Theorem 15.5 is the case where FC(G) = < 1 >. To deal with this case, we shall need an auxiliary result which allows us to reduce our study to a subgroup of finite index. For our convenience, we shall say that a just non-polycyclic group G with Fitt{G) * < 1 > is called primitive if the factor-group GIFitt{G) is primitive in the sense given in Chapter 8. 15.6. Lemma [RW] Let G be a just non-(polycyclic-by-finite) group with < 1 > * Fitt(G) periodic. Then G includes a subgroup H of finite index such that H is normal in HG and H has a primitive just non-polycyclic factor-group HIK. Proof We choose a section H\IK\ of G satisfying the following three conditions: (a)|G : Hx\ is finite; (b) H\IK\ is a just non-polycyclic group; (c) 0-rank of the Fitting factor-group of H\IK\ (that is O-rank of the group {H\IK\)IFitt{H\IK\)) is minimal under conditions (a) and (b). Let H2 = CoreG{H\). Then H2I(H2P\K\) is a finitely generated soluble-by-finite group which is not polycyclic and so by Lemma 15.1 H2/(H2 <~)Ki) has just non-(polycyclic-by-finite) images. Let H2IK2 be a just non-polycyclic group with K\ n H2 < K2. Since the Fitting factor-group of H2IK2 is clearly a homomorphic image of a Fitting factor-group of H2I{K\ f\ H2), the latter is isomorphic with a subgroup of H\ IK\. Consequently, the 0-ranks of the Fitting factor-group of H\IK\ and of H2IK2 are equal. Put L2IK2 = Fitt{H2/K2). By Lemma 8.1 the polycyclic-by-finite group H2IL2 includes a characteristic subgroup HIL2 such that the index \H2 : H\ is finite and HIL2 is a primitive polycyclic group. Thus H
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161

K2. Let K be a non-identity normal subgroup of W. If K fl Hy = < 1 > for some y e Y , then [KC\D,Hy] = < 1 >. Since F is transitive this implies that [ATID,D] = < 1 > so that Kf\D < CW{D) = < 1 >. This in turn implies that [K,D] = < 1 > and K < CW(D) = < 1 >. It follows that K n Hy * < 1 > for every y e Y, thus each Hyl(Hy fl K) is polycyclic-by-finite, and consequently WIK is polycyclic-by-finite too. 15.8. Proposition [RW] Let G be a group, H a subgroup of G, K a normal subgroup ofH. Write a for the canonical permutation representation ofG on the set of right cosets of H in G (so that Go = G/Coreo(H)). Let W be the unrestrictedpermutational wreath product: W = (HIK) WR (Go). Then there is a homomorphism y/ : G —* W satisfying Kery/ = Corea(K). Moreover, ifD is the base group of the wreath product then D(Gy) = W, Df] Gy/ = CoreG(Hyi), and the latter projects onto the canonical direct factors of the base group of (KCoreG(H)/K) WR Go. Proof Let Z be the set of right cosets of H in G and regard the base group of a wreath product X WR Go as the group of all mappings from S to X. We also write 8 for the "diagonal" mapping from .Yto this base group (so that if x e X, then x8 maps each element of £ to x); thus the subgroup < X5, Go > of X WR Go is the direct product of X8 and Go. A right transversal to H in G may be regarded as a function 1: 2 —• G mapping each coset to its representative, and so as an element of G WR Go. We form the mapping g - • i(g8)(go)rl from G to G WR Go; this

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is already a monomorphism and, as is observed in paper [CKK ], its image lies in H WR Ga. Thus we have an embedding q> : G —• H WR Ga. Define y/: G —• (H/K) WR Ga be the composite of cp and the canonical homomorphism £: H WR Ga — (H/K) WR Ga. Let g e G. Evidently gy/ lies in the base group D of (H/K) WR Ga if and only if g(p lies in the base group of H WR Ga, and from the definition of

, and so the mapping y/ will be an embedding. Moreover, since the index \G : H\ is finite, all wreath products which we are concern with are restricted wreath products. We have already choose the "good" section of G in Lemma 15.6. Thus we obtain 15.10. Proposition [RW] Let G be a just non-(polycyclic-by-finite) group. Then there is an embedding yi of G into the permutational wreath product W of a primitive just non-polycyclic group and a finite transitive permutation group such that D(Gy/) = W and D fl (Gy/) projects onto a subgroup offinite index in the canonical direct factors ofD, where D is the base group of W. Moreover, W itself becomes a just non-{polycyclic-by-finite) group.

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163

The last statement follows from Proposition 15.7. So we come to the next natural step. We will consider primitive just non-polycyclic groups. We have already obtained all needed facts in Chapter 8. 15.11. Theorem [RW] Let G be a primitive just non-polycyclic group with A = Fitt(G) * < 1 >,H = GIA. Suppose that A is periodic and £(H) =< 1 >. Then (i) A is an elementary abelian p - subgroup for some prime p; (if) A is a just infinite ¥PH - module; (Hi) H = P X T,P = CH(P) and P, T are free abelian subgroups offinite 0 rank; (iv) if U is a subgroup offinite index in T, then P <8>z Q is a simple QU module; (v) ¥PP - module A is torsion-free and has a finite rank. Proof By theorem 10.10 A is an elementary abelian p - subgroup for some prime p. Proposition 15.3 yields that^4 is a just infinite ¥PH - module. Theorem 10.10 gives also CH(A) = < 1 >. Theorem 8.5 shows that the statements (iii), (iv), (v) hold We finish with the following interesting remark. Let G be a group, H a normal subgroup of G. We say that H is nearly complemented in G or G is nearly splits over H if there is a subgroup X such that HP\X = < 1 > and the index \G : HX\ is finite. With this terminology, Theorem 15.4 asserts that if G is a just non-(polycyclic-by-finite) group with < 1 > * Fitt(G) torsion-free, then G nearly splits over Fitt(G). When A = Fitt(G) is periodic, we have only proved a partial version of this, namely G also nearly splits over Fitt(G) provided FC(GIA) is infinite. Actually, this is a fairly good result since the following result has been proved. 15.12. Theorem [RW] There exists a primitive just non-polycyclic group G which does not nearly split over its Fitting subgroup. The proof of the above result heavily depends on homological techniques exceeding the scope of this book, which is the reason for omitting it.

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Chapter 16 Just non-CC-Groups and Related Classes

Now we want to consider just non-Af-groups, where A" is a certain class of groups which includes both classes: the class T of all finite groups and the class A of all abelian groups. Specifically, we will consider the following candidates for X: central-by-finite groups, finite-by-abelian groups, FC-groups, and CC-groups. Just non-(central-by-finite) and just non-(finite-by-abelian) groups have been investigated in the remarkable paper of D. J. S. Robinson and Z. Zheng [RZ ]. While in [RW ] the key condition on the respectively group ring was to be noetherian, in the case of central-by-finite or finite-by-abelian group H the group ring Z// can not be noetherian. In this case the following fact is defining: the group H contains many almost central elements. Since central-by-finite and finite-by-abelian groups are contained in the wider class of FC - groups, the following natural step was the study of just non-FC-groups. This step has been realized by S. Franciosi, F. de Giovanni and L. A. Kurdachenko in [FdeGK 3]. In turn, the class of FC - groups is a subclass of the class of CC - groups. Therefore the study of just non-CC-groups, which has been initiated by L. A. Kurdachenko and J. Otal [KO 1, KO 2, KO 3], was a further natural prolongation of this research. This chapter is devoted to the main results of all mentioned above papers. As usual in order to avoid simple groups, we shall consider groups G with Fitt(G) * < 1 >. In this case, the first consequences of this assumption are collected in Chapter 10, with special mention to Theorem 10.5. The following lemma is crucial. 16.1. Lemma [KO 2] Let G be a group with a finite normal subgroup H such that GIH is a CC-group. Then G is a CC-group. In particular, a just non-CC-group has no non-identity finite normal subgroups. Proof

Let g e G. Put LIH = < gH >GIH . We want to show that GICG(gG) is a

165

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Just non-X-Groups

Chernikov group, so we only need to prove that GICG(L) is a Chernikov group. Since GIH is a CC-group, GICG{LIH) is a Chernikov group. Moreover, either LIH is a Chernikov subgroup or LIH includes a normal Chernikov subgroup TIH such that LIT is infinite cyclic [PY]. On the other hand, H is finite and so is GICG(H). Put K= Lr\CG(H).Then LIK is finite and KC\H in particular, GICG(KI(K n #)) is a Chernikov group. Thus so is G/C where C = CG{Kf)H) n CG(K/(KnH)). Since C stabilizes the chain < 1 > < Kf]H< K and ATD/^ is central in £, there exists the mapping 0 : C —>• Hom(K,KDH) given by 0(c) (JC) = [c,x], where c e C and x e AT with Ker9 = CG(AT) [KW, Chapter I, C]. Let « = \H\. Given C E C since [c,x] e / / for all x <E K, we have 1 = [c,x]" = [c,x"]. Thus A"" < KerQ and 0 induces the canonical homomorphism q> : C —• Hom(KIKn,KV\H) given by . We note that GICG(K\) is a Chernikov group since GICG{K) is and .Ki < K. The group Z/Ki includes a finite normal subgroup 77ATi giving an infinite cyclic factor-group. If C\ = CG(TIK\) fl CG(LIT), GIC\ is finite and C\ stabilizes the chain < 1 > < TIK\ < LIKy, so Proposition I.C.3 of book [KW] assures that C\ICG{LIK\) is finite, thus GICG(LIK\) is finite too. Put Ci = CG{LIK\) fl CG(K\), so that GICi is a Chernikov group. Once again, applying Proposition I.C.3 of [KW] to the chain < 1 > < K\ < L we can write CjICdiL) < Xx!ExEx and Ex = t^(K\) for all x e X. Since K\ is a Chernikov subgroup and X is finite, CilCciiL) is a Chernikov group. Then GICd{L) and G/CG(L) are Chernikov groups, as required. The last assertion is obvious. 16.2. Theorem [KO 1] Let G be a group including a normal infinite cyclic subgroup. Then G is a just non-CC-group if and only if the following conditions hold: (/) C, (G) is a non-identity torsion-free locally cyclic subgroup; (ii) Glt^iG) is a torsion-free abelian group; (Hi) for every x the subgroup [G,x] is minimax. Proof Let G be a just non-CC-group and suppose that C = < c > is an infinite normal subgroup of G. Since G/C is a CC-group, its elements of finite order form a characteristic subgroup TIC = t(GIC) [PY]. Put H = CG(Q; then \G : H\ < 2, in particular, GIH is cyclic. Since TC\H is central-by-(locally finite), [TC\H,TC\H] is locally finite (see, for example, [RD 9, Corollary to Theorem 4.12]). So the set T\ of all elements of finite order of Tf] H is a characteristic subgroup and (7T) H)ITi is torsion-free abelian. Since T\ fl C = < 1 > , Lemma

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167

10.3 yields that T\ = < 1 >. Thus Z = TflH is a torsion-free abelian locally cyclic subgroup. Let z e Z. Since Z/C is periodic, zr e C for some r e N. If A e / / and z\ = z \ then z{ = (zh)r = (xr)h = zr, and so zx = z, that is Z < £(//). Suppose that # * G. Thus G = / / < c? > and c/2 e # < CG(Z), it follows that af inverts any element of Z On the other hand, we recall that [G, G] < Z. Thus if k f f , then [d,h~l] = d~lhdh~l = z e Z, that is ( f ' M = zA, and so fi?-2/!^2 = d-l(d-lhd)d = of-'zM = {d~lzd){d^hd) = z-'zA = A. Therefore d1 e CG(H) and then d2 e f(G). Lemma 10.3 yields that in this case d2 = 1. Thus < dZ > f]H/Z = < 1 > and G/Z =< dZ > xH/Z. If there is some a e £(H)\Z, then the subgroup < a,Z > is normal in G since [G, G] < Z. If some a" e Z, the element aZ would have finite order in the torsion-free group HIZ, which would imply a e Z, which is impossible. Hence < a > C\Z = < 1 > and < a,Z > = < a > xZ By Theorem 2.7 of the paper [FdeGK 2] < a,Z> includes a non-identity G-invariant subgroup M such that Mfl Z = < 1 >, contradicting Lemma 10.3. This implies that Z = £(//). Now let h e H\Z. By the above conclusion, there exists some y e H such that 1 * [h,y] = z\ e Z. Put [h,d] = z2 s Z. Since Z is locally cyclic, there exists u e Z such that < zi,Z2 > = < « > . We can find that < h > fl < w> = < 1>, so < h,u > = < h > x < u >. Therefore /!•>' = /i«' for some t =t 0. Similarly, /zrf = /JU*. Furthermore, ofyof"1 = yzj, with Z3 e Z, and then dy - yz->,d = z-^yd. It follows that }& = hz^d = A^. But /i* = (AM*)* = hu'+k, while A*d = Aw*"'. Since d inverts each element of Z and HIZ is torsion-free, this implies that t = 0, a final contradiction, which shows that H' = G and hence Z = C(ff)- This satisfies (i) and (ii). Let x e G. If 1 * v e £(G) and V = < v >; then K is an infinite cyclic subgroup and V is normal in G. Since G/F is a CC-group, [GIV,xV\ is a Chernikov subgroup and so [G,x] is minimax, that is G also satisfies (iii). Conversely, suppose that G is a group satisfying the conditions (i) - (iii). Since G is torsion-free and non-abelian, G is not a CC - group. Let H be a non-identity normal subgroup of G. Givenx e G\£(G) and put W = [G,x]. We note that Wis a minimax subgroup and W < C(G). Since G is a nilpotent group, M = H{M^{G) * < 1 >. Moreover MOW = < 1 >; otherwise < M,W> = MxW< f(G), contradicting (1). Therefore R = Mfl ^ * < 1 >. Now r0(W) = r0(R) so that WIR is a periodic factor-group of a minimax group. It follows that [G/R,xR] = [G,x]/W? = WIR is a Chernikov group. Since R < H, [GIH,xH] is a Chernikov group too and it readily follows that GIH is a CC-group. Since G is nilpotent of class 2, G/CG(X) S [G,X] for every x e G. It follows that G/Ca(x) is minimax by condition (iii). In other words, G Aos minimax conjugacy classes.

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16.3. Corollary [FdeGK 3] Let G be a group with a non-identity center. Then G is a just non-FC-group if and only if the following conditions hold: (i) ((G) is a torsion-free locally cyclic subgroup; (ii) GI((G) is a torsion-free abelian group; (Hi) for every x the subgroup [G,x] is cyclic. Proof Since every just non-FC-group is also a just non-CC-group, G satisfies (i) and (ii) by Theorem 16.2. Let x e G. If x e ((G), then [G,x] =< 1 >. Let x <£ ((G), 1 * z e ((G). As Gl < z > is an FC-group, the subgroup < x,z >G I < z > satisfies Max, so that < x >G has the same properties. In particular [G,x] is finitely generated and hence it is infinite cyclic. The proof of sufficiency is similar to the proof of sufficiency of Theorem 16.2. 16.4. Corollary [RZ] Let G be a group with non-identity center. Then G is a just non-(finite-by-abeliari) group if and only if the following conditions hold: (i) ((G) is a torsion-free locally cyclic subgroup; (it) GI((G) is a torsion-free abelian group; (Hi) [G, G] is infinite cyclic. Proof By Theorem 16.2 G satisfies (i) and (ii). Let l * z e [G,G]. Since Gl < z > is finite-by-abelian, [Gl < z >, Gl < z >] is finite. It follows that [G, G] is cyclic-by-finite. Since [G, G] is locally cyclic, [G, G] is infinite cyclic. Conversely, let G be a group satisfying conditions (i) - (iii), H a non-identity normal subgroup of G. Then Hn ((G) * < 1 >. Since [G,G] < ((G) and ((G) is locally cyclic, H D [G, G] * < 1 >. It follows that GIH is finite-by-abelian. These above results describe in a satisfactory way the case ((G) * < 1 >. In the sequel we study the more difficult complementary case in which ((G) = < 1 >. As we mentioned above in this case the usual condition Fitt(G) * < 1 > is assumed. 16.5. Theorem [KO 2] Let G be a just non-CC-group. IfFC(G) ((G) = < 1 >, then G is a just non-Chernikov group.

±<\>but

Proof Let 1 * x e FC(G),X = < x >G . Then \G : CG(X)\ is finite and X is central-by-finite. By Schur's theorem (see, for example, [RD 9, Theorem 4.12]) [X,X] is finite. Lemma 16.1 yields that [X,X\ = < 1 >. In other words, X is a finitely generated abelian subgroup. The periodic part of X is a finite G-invariant subgroup, so Lemma 16.1 implies that Jf is torsion-free. Put C = CG(X). We can assume that X is Z-irreducible. Since GIX is a CC-group, all its elements of finite order form a subgroup TIX [PY]. Put Tx = T(\ C. Then [TX,T{[ is periodic (see, for example, [RD 9, Corollary to Theorem 4.12]). In particular, [TuTi] nX = < 1 >. Lemma 10.3 implies that [T\,T\] = < 1 >. The same arguments show that

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169

T\ is torsion-free abelian subgroup. Since T\IX is periodic, then T\ has finite 0 -rank. It is easy to see that CG(T\) = CGVO = C. Since the periodic part of a CC-group includes the derived subgroup [PY], the factor-group (C/X)/(Ti/X) = CITy is abelian. It follows that CIT\ < {(G/Ti), because CITi n[G/TuG/Ti] = < 1 >. Suppose that C * T\. Let LIT\ be a non-identity locally cyclic subgroup of CIT\. Since T\ < £(Z,), L is abelian. Since CIT\ < £(G/Ti), L is normal in G. Suppose that T * T\. By Theorem 2.7 of the paper [FdeGK 2] there exists a non-identity ^-invariant subgroup Q < L such that Q fl T\ = < 1 >, because TICT(L) is finite. The inclusions [L,T\ < Tx and [0,7] < Q imply [Q,T\ = < 1 >, in particular, CL(T) * < 1 >. By the choice of X we have CL(T)V\X= < 1 >. This contradicts Lemma 10.3, since the subgroup CiiT) is G - invariant. Thus T= Ti. In other words, T < C and GIC is abelian and torsion-free. Then GICGU-) is abelian. Put U = I ® z Q / = r ® z Q . The application of Theorem 1' of the paper [ZD 2] gives the decomposition U = V (B W where W is a Q - submodule such that G = CG(W)- Since U is an essential extension of L, WPi L * < 1 >. But this means that £(G) * < 1 >. This contradiction proves that C = T. For y e G\C put / / / I = C G/X (< y >° XIX). Let E = Hp\C and consider the mapping 6 : E —* E given by ed = [e,y], e € E. Clearly 6 is a Z / / endomorphism of E with ATerfl = Cf(y) and Imd = [E,y]. Since [E,y] < X, EICEiy) is a torsion-free finitely generated abelian group. If 6 is not injective and Ei = C^(y) we note that E\ *< 1 > and that £ i has only finitely many conjugates in G, say {Ef,... ,£Jf")-. Put Ej = Ef fl... fl£f"; then by Remak's theorem E/E2 <E/Ef

x... x £ / £ f .

Since £ / £ f = Egl/Ef = EIE\, ElEi is a torsion-free finitely generated abelian group. Since X and also C are Z-irreducible, £2 = < 1 >• It follows that E is finitely generated. If 9 is injective then E = [£,.y] < X, and so it is clear that E is finitely generated in this case. Since X * < 1 >, GIX is a CC-group, so GIH is a Cheraikov group and CIE is a Chernikov group too. All these facts imply that C is a minimax subgroup. Let U be a non-identity normal subgroup of G. By Lemma 10.3 t/ fl X * < 1 >. Since a periodic minimax group is a Chernikov group, CI(U(~\X) is a Chernikov group. It follows that GIU is a Chernikov group. In other words, G is a just non-Chernikov group. The above result naturally raises the question about the structure of just non-CC-groups with an identity FC-center. As in other cases of just non-A"-groups, the strategy will consist in splitting into two complementary cases:

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the non-monolithic case and the monolithic case. 16.6. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = FittiG) * < 1 > and suppose that A is not torsion-free. Ifa\,... ,a„ e A,B = < a\ >G ... < a„ >G , then B is a just infinite ZH - module where H = GIA. Proof Let T be the periodic part of A. If FC(G) * < 1 >, then Lemma 10.3 yields that TTl FC(G) *• < 1 >. But in this case G includes a non-identity finite normal subgroup, which contradicts Lemma 16.1. Thus FC(G) = < 1 >. Corollary 10.6 imply that A is an elementary abelian p-subgroup for some prime p. Let M = {C | C is a non-identity G-invariant subgroup of B}. Since G is a non-monolithic group, M. * 0, and we can choose C e M.. Since GIC is a CC-group, < a, > G CIC is a Chernikov group [PY], so it is finite. It follows that BIC is finite. Since G is a non-monolithic group, (~\M = < 1 >, hence, 5 is a just infinite Z / / - module. 16.7. Lemma [KO 1] Let G be a non-monolithic just non-CC-group with FC{G) = < 1 >, A = Fitt{G) * < 1 >, ai,... ,a„ e A, B = < a\ >G ... < a„ >G ,H = GIA. Suppose that A is torsion-free. If the ZH module A is Z-irreducible, then B is a just infinite ZH - module. Proof

As above, put M = {C | C is a non-identity G-invariant subgroup of B}.

Again it is sufficient to prove that BIC is finite for any C e M. Since GIC is a CC-group, < a, > G CIC is a Chernikov subgroup because AIC is periodic. Since A is abelian, < a, >G CIC is a bounded group, so that < a, > G CIC is finite. Hence and BIC is finite. 16.8. Lemma [KO 1] Let G be a non-monolithic just non-CC-group with FC(G) = < 1 >, A = Fitt(G) * < 1 >, ai,... ,a„e^, 5 = < ai > G ... < a„ >G ,H = GIA. Suppose that A is torsion-free. Then B includes a G-invariant subgroup C * < 1 > such that BIC is Chernikov-by-polycyclic, and ZH - module C is just infinite. Proof Let £ be a non-identity G-invariant subgroup of B. Since GIE is CC-group then < a, > G EIE includes a G-invariant Chernikov subgroup EJE such that (< a, > G E/E)/Et is cyclic [PY]. Then BIE includes a G-invariant Chernikov subgroup FIE = (E\... E„)IE such that BIF is finitely generated,

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moreover ro(B/F) < n. We note also that ro(B/E) = ro(B/F). Let [/be a non-identity G-invariant subgroup of B with ro(B/U) is maximal. Since GIU is a CC-group, we can obtain that BIU again includes a G-invariant Chernikov subgroup VIU such that BIV is finitely generated and torsion-free. Let W be a non-identity G-invariant subgroup of V. By the choice of V it follows that r0(BIU) = rQ{BIW). In particular, n(UIW) = 0, in other words, £//Jf is periodic. This means that a Z//-module t/ is Z- irreducible. Let Y be a non-empty finite subset of U, C = < F > G .By Lemma 16.7 the ZH - module C is just infinite. By above BIC is Chernikov-by-polycyclic. Lemmas 16.6, 16.7, 16.8 show the role of just infinite modules in the study of just non-CC-group. Now we can use results of Chapters 6 and 7. 16.9. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, and suppose that A is an elementary abelian p-subgroup for some prime p, H = GIA. Then Op(H) = < 1 >. Proof Suppose the contrary. Then there exists a normal subgroup P of G such that P * A and PI A is a finite p-group. By Corollary 10.6 A = Ca(A). Therefore there exists an element a e A such that P % CG{O). Thus if B - < a >G , then P $ CG(B). Since CG(B) > A, this gives that PCG(B)/CG(B) is a non-identity finite normalp - group. Hence OP(GICG(B)) * < 1 >. Lemma 16.6 yields that B is a just infinite ZH - module. Theorem 6.15 implies that in this case OP(GICG(B))

= < 1 >. We arrive to a contradiction.

16.10. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >,FC{G) = < 1 >,H = GIA. Suppose that A is an elementary abelian p-subgroup for some prime p. Then Opi(H) is finite. Proof Suppose that Opi{H) is infinite. Since G is non-monolithic, A includes a proper non-identity G-invariant subgroup B. Let a € A\B. Since GIB is a CC-group and aP = \,G BIB = CIB * < 1 > is finite. Let Q = CH(CIB), so that HIQ is finite. In particular, Q fl Opi(H) * < l >. It follows that Q f) Opi(H) includes a non-identity finite G-invariant subgroup L. Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]) implies that C = Bx D for some L-invariant subgroup D. If x e L, then [D,x] < D and since x centralizes CIB,[C,x] < B. This gives [D,x] , so CA(L) * < 1 >. By Maschke's theorem we have the direct decomposition A = CA(L) XE where £ is a /.-invariant subgroup. It follows that E < [A,L]. Since L is normal in H, CA{L) and [A,L] are G-invariant subgroups of A. They intersect by identity, so that [A,L] = < 1 >. Corollary 10.6 implies that L < CG{A) = A. This is a contradiction.

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16.11. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) =<\>,H= GIA. Suppose that A is an elementary abelian p-subgroup for some prime p.IfT is a periodic normal locally soluble subgroup ofH, then T is finite. In fact, by Lemma 16.9 Op(T) = < 1 >, and Lemma 16.10 proved that Opi(T) is finite. Lemma 14.13 implies the finiteness of T. 16.12. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) = < 1 >, H = GIA. Suppose that A is an elementary abelian p-subgroup for some prime p. Then H is an FC-group. Proof Given an x e H, we put X = < x >H . Then X is Chernikov-by-cyclic [PY]. If D is the divisible part of X, then D is a periodic divisible abelian normal subgroup of//. By Corollary 16.11 D has to be finite and so identity. Hence X is finite-by-cyclic and therefore H is an FC-group. Now is the turn of the case when Fitt{G) is torsion-free. 16.13. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >, FC(G) = < 1 >, H = GIA. Suppose that A is torsion-free abelian. If A includes a non-identity G-invariant subgroup B such that B is Z-irreducible as a ZH - module, then the ZH - module A is also Z-irreducible. Proof It clearly suffices to show that AIB is periodic. Suppose that it is false and choose an element aB of AIB such that \aB\ is infinite. Since GIB is a CC-group, EIB =< a >G BIB includes a G-invariant Chernikov subgroup UIB such that Ell] is cyclic [PY]. In particular, the ZH - module U is Z-irreducible. Since E/U is infinite cyclic, \G : CG(U/B)\ < 2 and UIB is Z-irreducible too. Since FC{G) = < 1 >, we have that G/CG(B) is infinite. Corollary 4.4 yields that E includes a G-invariant subgroup C * < 1 > such that UC\ C = < 1 >. But this contradicts Lemma 10.3. 16.14. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) = < 1 >,H = GIA , 1 * a e A,B = < a >G . Suppose that A is torsion-free abelian. Then B is a just infinite "LH - module and AIB is periodic. In particular, the ZH- module A is Z-irreducible. Proof Corollary 16.8 implies that B includes a G-invariant subgroup E such that BIE is Chemikov-by-polycyclic and E is a just infinite Z//-module. Let TIE be the periodic part of BIE; then the Z//-module T is Z-irreducible. By Lemma 16.13 the ZH- module A is Z-irreducible. Lemma 16.7 yields that B is a just infinite ZH-

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module. 16.15. Corollary [KO 1] Let Gbe a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC(G) = ,H=G/A,l±aeA,B=G.IfA torsion-free abelian, then CG(B) = A.

is

Proof Corollary 16.14 yields that AIB is periodic. Let b e A, g e CG(B), b\ = bg. There is a number n e N such that b" e B. We have now b\ = {b%y = (b")g = b". It follows that b\ = b since A is torsion-free. Hence CG(B) = CG(A) = A. 16.16. Corollary [KO 1] Let Gbe a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >,FC(G) = < 1 >,H = GIA. If A is torsion-free abelian, then H is an FC - group. Proof Let 1 * a e A,B = < a >G . Corollary 16.14 shows that B is a just infinite ZH - module. By Corollary 16.15 CG(B) = A. Put M. = {E | E is a non-identity G-invariant subgroup of B}. Let x e H,X = < x >H . Since His a CC-group, we can observe that either Xis a Chernikov subgroup or X includes a normal Chernikov subgroup Y such thatX/7 is infinite cyclic [PY]. Let D be the divisible part of X. If E e M, then B/E is finite, so that \H : CH(BIE)\ is finite. It follows that D n CH(BIE) = D so D < CH{BIE) or [B,D] < E. Since it is valid for every E e M, then [B,D] < f] M = < 1 >. In other words, either X is finite or finite-by-cyclic. Consequently, H is an FC-group. 16.17. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) =,H = GIA. If A is torsion-free abelian, then H is central-by-finite and almost torsion-free. Proof Let 1 # a e A,B = < a >G . Corollary 16.14 shows that B is a just infinite ZH - module. By Corollary 16.15 CG(B) = A. Corollary 16.16 yields that H is an FC-group. Therefore it suffices to use Corollary 6.16. 16.18. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >,FC{G) = < 1 >, \ ± a <^ A,B = < a >a . Suppose that A is torsion-free abelian. IfA^zAe C,(GIA) and U = C G (< z >G BIB), then Uf\A is a just infinite ZG - module. Proof Firstly we note that A = CG(A) by Corollary 10.6. Since zA e C,(GIA), the mapping 6 : a —* [a,z], a e A, is a ZG - homomorphism, in particular,

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Kerd = CA{Z) and ImQ = [A,z] are G-invariant subgroups of A. It is easy to see that CA(Z) is a pure subgroup of A. Corollary 16.14 shows that CA(Z) = < 1 >, i.e. 9 is a monomorphism. In other words, A and [A,z] are G-isomorphic; similarly Uf)A =zo [Ur\A,z]. From the choice of z it follows that [Uf]A,z] < B. This means that [UC\A,z] is a just infinite ZG - module, hence and UnA is a just infinite ZG - module too. 16.19. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >,FC(G) = < 1 >. If A is torsion-free abelian, then A is just infinite ZG - module. Proof Corollary 10.6 gives the equality A = CG(A). Corollary 16.17 implies that GIA is central-by-finite and almost torsion-free. Let 1 ±aeA,B=G , A ± zA e C(GA4) and \zA\ is infinite, UIB = CG/B(< z >G BIB). Lemma 16.18 yields that Iff) A is a just infinite ZG-module. In particular, {UC\A)IB is finite. It follows that UIB is an FC - group, since U/(Uf]A) = UA/A < GIA and GIA is central-by-finite. Since UIB is not periodic, £(£///?) * < 1 >; moreover, (UIB)lt;(yiB) is periodic (see, for example, [RD 9, Theorem 4.32]). By Corollary 16.17 U/(Up\A) is central-by-finite and almost torsion-free, in particular, the periodic part 77(1/n A) of U/(Uf) A) is finite. Since (U n A)IB is finite, TIB is the periodic part of UIB. Let ZIB = {(UIB); then Z/B = (Z/B n 775) x X/B (see, for example, [FL 1, Theorem 27.5]). Let / = \ZIB n TIB\; then {ZIB)' = 175 < X/5, in particular, YIB is a non-identity torsion-free normal subgroup of GIB. Since the derived subgroup of CC-group is periodic [PY], YIB < C,{GIB). Let B±yB e YIB; then C G /B(VB) = GIB. By Lemma 16.18 ^ is a just infinite TLG module. 16.20. Lemma [FdeGK 3] Let G be a non-monolithic just non-FC-group, A = Fitt{G) * < 1 >, H = GIA, FC(G) = < 1 >. Suppose that A is an elementary abelian p-subgroup for some prime p. IfG is not periodic then A is a just infinite ¥PH- module. Proof We note that, by Corollary 10.6 H can not be periodic. In this case its center f (//) contains an element z of infinite order (see, for example,[RD 9, Theorem 4.32]). The mapping q>: a —• [a,z], a e A, is a G-endomorphism of A. Again Im

, then GICA(Z) is an FC-group. It follows that AICA(Z) includes a non-identity finite G-invariant subgroup UICA(Z). Since U $ Ker = [U,z] * < 1 >. From the G isomorphism [A,z] = 7m

. Let 1 * a e A,B = < a >G ,C/B = CA/B(z). Then B is a just infinite ¥PH module by Lemma 16.6. Since GIB is an FC-group, AIC is finite. As z e £(//), C

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is an ¥PH - submodule of A. Clearly Kercp f] C = Cc(z) = < 1 >, so that

= [C,z] are G-isomorphic. Since [C,z] < B, then the WPH - submodule [C,z] is just infinite. It follows that and C is a just infinite ¥PHsubmodule. Thus CIB is finite and then AIB is finite. Therefore A is a just infinite ¥PH - module. In the next results we are developing the basic features of the non-monolithic case. 16.21. Theorem [KO 1] Let G be a non-monolithic just non-CC-group with C(G) = < 1 > andFitt(G) * < 1 >. (1) IfFitt(G) = A is a non-torsion-free subgroup, then G is a just-non-FC-group. (2) IfFitt(G) = A is torsion-free then either G is a just non-FC-group or G is a just non-Chernikov group. Proof (1) Corollary 10.6 yields that A is an elementary abelian/? - subgroup for some prime p. By Corollary 16.12 GIA is an FC-group. Let U be a non-identity normal subgroup of G,V=ADU. By lemma 10.3, V*• < 1 >. Let x e G\V,XIV = < x >G VIV. Since GIV is a CC-group, XIV includes a normal Chernikov subgroup YIV such that XIY is cyclic [PY]. Since A is an elementary abelian then (XIV) fl (AIV) is finite. Since GIA is an FC-group, XAIA is finite or finite-by-cyclic. It follows that XIV is finite or finite-by-cyclic too. Hence GIV, and therefore G/U, is an FC-group. (2) Corollary 10.6 yields that ,4 is a torsion-free abelian subgroup. Suppose that FC(G) = < 1 >. By Corollary 16.17 H= GIA is central-by-finite and Corollary 16.19 implies that A is just infinite Z77 - module. Let again U be a non-identity normal subgroup of G, V = A fl U. By Lemma 10.3, V * < 1 >. Theny4/Kis finite and therefore GIV is a finite-by-FC-group, i.e. GIV is also FC-group. Hence GIU is also an FC - group. If G is an FC-hypercentral group, then G is a just-non-Chernikov group by Lemmal6.5 16.22. Corollary Let G be a non-monolithic just non-CC-group with f(G) = < 1 > and Fitt(G) = A * < 1 >. Suppose that A is an elementary abelian p subgroup for some prime p. IfG is locally soluble, then (1) A is a just infinite ¥PH - module where H = GIA ; (2) H is central-by-finite and almost torsion-free ; (3) every proper factor-group ofG has a finite derived subgroup. Proof By Corollary 10.6 A = CG(A). Suppose that H is periodic. Corollary 16.11 shows that in this case H is finite. It follows that A includes a non-identity finite G - invariant subgroup, what contradicts to Lemma 16.1. Hence H is not periodic. Lemma 16.20 implies that A is a just infinite ¥PH - module. Finally,

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Corollary 6.19 shows that H is central-by-finite and almost torsion-free. Let U be a non-identity normal subgroup of G, V = A f) U. By Lemma 10.3 V * < 1 >. Since A is just infinite then AIV is finite. It follows that GIV is finite-by-(central-by-finite). By Schur's theorem, a central-by-finite group has a finite derived subgroup (see, for example [RD 9, Theorem 4.12]). Therefore, GIV and hence G/U has a finite derived subgroup. 16.23. Corollary Let G be a non-monolithic just non-CC-group with C(G) = < 1 > and Fitt(G) = A * < 1 >. Suppose that G is not a just-non-Chernikov group and A is torsion-free. Then (1) A is a just infinite ZH- module where H = GIA ; (2) H is central-by-finite and almost torsion-free; (3) every proper factor-group ofG has a finite derived subgroup. Proof Again A = CG(A) is assured by Corollary 10.6. Since G is not just non-Chernikov, then FC{G) = < 1 >. Corollary 16.17 yields that H is central-by-finite and almost torsion-free, and Corollary 16.19 implies that A is a just infinite ZH - module. The proof of (3) is the same as in Corollary 16.22. 16.24. Theorem [RZ] Let G be a non-monolithic just non-(finite-by-abelian) group with £(G) = < 1 > and Fitt{G) * < 1 >. Then there is a subgroup X such thatXHA = < 1 > and \G : XA\ is finite. In other words, G nearly splits over A. Proof Clearly G is a just non-FC-group, therefore Corollaryl0.7 yields that either A is an elementary abelian p - subgroup for some prime p or A is torsion-free and abelian. In both these cases CG(A) -A. Since GIA has a finite derived subgroup, it is nilpotent-by-finite. Let A be elementary abelian. If we assume that GIA is periodic, then Corollary 16.11 yields that GIA is finite, what is impossible. Thus GIA has elements of infinite orders. By Lemma 16.2 A is just infinite ¥PH - module where H = GIA. Corollary 6.19 shows that H is central-by-finite and almost torsion-free. If A is torsion-free, then A is a just infinite ZH - module by Corollary 16.23 and also H is central-by-finite and almost torsion-free as well. Let 1 * zA £ £(G/A). The mapping , so cp is injective. Since Im

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Then for every g e G one has [g,z] = cij[uj,z] for some Uj e A,j e {[,... ,m}. Hence [guj\z] = aj = [gj,z] and gw/'g"' e X, so that G = < gu... ,gm,XA >. Now |.¥[G,G]^ : XA\ is finite and G/[G,G]XA is finite, being a finitely generated abelian periodic group. Hence \G : XA\ is finite. Finally, XC\A = CA(Z) = < 1 >, so the proof is complete. 16.25. Corollary [RZ] Let G be a non-monolithic just non-(finite-by-abeliari) group with f (G) = < 1 > and Fitt(G) = A *• < 1 >. Then every proper factor-group ofG is central-by-finite. Proof. By Corollary 10.7 A = CG{A). As in Theorem 16.24 H = GIA is central-by-finite and A is a just infinite J,H - module. Let X be a subgroup such that/1 f l l = < 1 > and \G : XA\ is finite. This subgroup exists by Theorem 16.24. In particular, X i s central-by-finite. Put Z = C,{X). Let U be a non-identity normal subgroup of G, V = A f] U. By Lemma 10.3 V * < 1 >. Since A is a just infinite ZH - module, AlV is finite and A/VDXVIV = < 1 >. It follows that the index \AXIV : XVIV\ = \A/V : XVIV\ = \AIV\ is finite. Furthermore, the index \GIV : XVIV\ = \G :AX\ is finite. Since \X : 2\ is finite then GIV includes the abelian subgroup ZVIV of finite index. It follows that G/Vis central-by-finite (see, for example [TM 1, Lemma 7.5]). Therefore GIU'xs, also central-by-finite. 16.26. Corollary Let G be a non-monolithic just non-CC-group with C, (G) = < 1 > andFittiG) = A * < 1 >. (1) If A is periodic and G is locally soluble, then every proper factor-group of G is central-by-finite. (2) If A is not periodic, then either G is a just non-Chernikov group or every proper factor-group ofG is central-by-finite. 16.27. Corollary [FdeGK 3] Let G be a non-monolithic just-non-FC-group with Z(G) = < 1 > andFitt{G) = A * < 1 >. (1) If A is periodic and G is locally soluble, then every proper factor-group of G is central-by-finite. (2) If A is not periodic, then every proper factor-group of G is central-by-finite. 16.28. Theorem [RZ] (1) Let G be a non-monolithic just-non-(central-by-finite) group with f(G) = < 1 > andFitt{G) = A i= < 1 >. Then A is just infinite ZH- module where H = GIA is central-by-finite and almost torsion-free and CH(A) = < 1 >. Moreover, there exists an abelian torsion-free subgroup Zsuch that Zf]A =< 1 > and the index \G : AZ\ is finite. (2) Conversely, let H be a central-by-finite and almost torsion-free group, A a just infinite IJH - module such that CH(A) = < 1 >. Then every extension of A by

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H including the given module structure is a non-monolithic just non-(centralby-finite) group with the identity center and the Fitting subgroup A. Proof (1) As in Corollary 16.19 and Lemma 16.20 we can prove that A = Ca(A) is a just infinite ZH - module where H = G/A is central-by-finite and almost torsion-free. By Theorem 16.24 there exists a subgroup X such that XC\A = < 1 > and \G : AX\ is finite, in particular, X is central-by-finite and almost torsion-free. Therefore £(X) includes a torsion-free subgroup Z such that the index \X : Z\ is finite. Hence and index \G : AZ\ is also finite. (2) Let G be an extension of A by H, U a non-identity normal subgroup of G, V = AC\U. Since CG{A) = A, then V± < 1 >. The factor-group GIV is an extension of the finite subgroup AIV by the finite-by-abelian group G/A, hence G/Vhas a finite derived subgroup. Thus and G/U has a finite derived subgroup. Clearly, G is non-monolithic. If we assume that £(G) * < 1 > then £(G) C\ A •*• < 1 >. However it contradicts Corollary 6.5. Finally, by Corollary 10.7 the subgroup Fitt(G) is abelian and includes A, so that A = Fitt(G). Now we can use Corollary 16.25. In order to complete our study of just non-CC-groups, we only need to consider the monolithic case. To deal with it, we can use some results from Chapter 3. 16.29. Lemma [KO 3] Let G be a monolithic just non-CC-group with £(G) = < 1 > andFitt{G) * < 1 >. Then Fitt(G) is the monolith ofG. Proof Let M be the monolith of G; then M is abelian. Suppose that M * Fitt{G) = A.VxxXH = G/A, then H is a CC - group. We regard A as a ZH module, because A is abelian by Corollary 10.6. In this notations M is a simple 7LH - submodule of A. By Corollary 10.6 either A is an elementary abelian p-subgroup for some prime pox A is a torsion-free abelian subgroup. Suppose that A is an elementary abelian p-subgroup. Since GIM is a CC - group, AIM includes a non-identity G - invariant finite subgroup. In particular, AIM includes a minimal finite G - invariant subgroup B/M. By Corollary 4.3 there is a G - invariant non-identity subgroup C such that B = M x C, in particular, M f l C = < 1 >. But this contradicts Lemma 10.3. If A is torsion-free, it turns out that M is also divisible. In particular, A = M x D, for some subgroup D (see, for example, [FL 1, Theorem 21.2]). It follows that AIM is torsion-free and in this case AIM includes a non-identity G-invariant subgroup VIM which is infinite cyclic. By Corollary 4.4 U must include a non-identity G - invariant subgroup Fsuch that M R V = < 1 >, which again leads to a contradiction. Consequently, M = A. 16.30. Theorem [KO 3] Let G be a monolithic just non-CC-group with f(G) = < 1 > andFitt{G) = A * < 1 >. Suppose that A is not torsion-free. Then

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(i) there is a prime p such that A is an elementary abelian p-subgroup; (ii) A is the unique minimal normal subgroup ofG; (in) A = CG(4); (iv) if H = GIA and S = Socab{H), then S is a pi-subgroup including a subgroup R such that SIR is locally cyclic and Coren(R) = < 1 >; (v)ifFitt(H) * < 1 >, then G splits conjugately over A. Proof By Corollary 10.6 A is an elementary abelian p - subgroup for some prime p and CG(A) = A. Lemma 16.29 yields that A is the monolith of G. From Theorem 3.10 we obtain (iv). Finally, let LIA = Fitt(GIA). Suppose that £(L) * < 1 >. Then A < £(L) , so L < Fitt(G) = A. This contradiction shows that C,(L) = < 1 >. Theorem 4.5 proves now (v). 16.31. Theorem [KO 3] Let G be a monolithic just non-CC-group -with C, (G) = < 1 > andFitt(G) = A * < 1 >. Suppose that A is torsion-free. Then (i) A is a divisible torsion-free abelian subgroup; (ii) A is the unique minimal normal subgroup ofG; (Hi) A = CG(A); (iv) ifH = GIA andS = Socab(H), then S includes a subgroup R such that SIR is locally cyclic and CoreH(R) = < 1 >; (v) G splits conjugately over A. Proof By Corollary 10.6 A is torsion-free abelian and A = CQ(A). Lemma 16.29 yields that A is the monolith of G. In particular, A is a minimal normal subgroup of G, so that A is a divisible subgroup. From Theorem 3.10 we obtain (iv), and Theorem 4.5 gives (v). One more time, as an easy consequence of the above theorems, we obtain the following results. 16.32. Corollary [FdeGK 3] Let G be a monolithic just non-FC-group with C(G) = < 1 > andFitt(G) = A *• < 1 >. Suppose that G is locally soluble. Then (i) A is the unique minimal normal subgroup ofG; (ii)A = CG(A); (iii)G splits conjugately over A; (iv) if H = GIA, S = SocH, then S includes a subgroup R such that SIR is locally cyclic and CoreH(R) = < 1 >. Moreover, ifA is an elementary abelian p subgroup, then S is a p1-subgroup. 16.33. Corollary Let G be a monolithic just non-(finite-by-abelian) group with C, (G) = < 1 > andFitt(G) = A * < 1 >. Suppose that G is locally soluble. Then (i) A is the unique minimal normal subgroup ofG;

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(ii)A = CG(A); (Hi) G splits conjugately over A; (iv) if H = GIA, S = SocH, then S includes a subgroup R such that SIR is locally cyclic and Core H(R) = < 1 >. Moreover, ifA is an elementary abelianp subgroup for some prime p, then S is a pi- subgroup. If G satisfies the hypothesis of Corollary 16.33, then the derived subgroup KIA ofH= GIA is finite. As a consequence, CIA = CGIA(K/A) has finite index in GIA and CIA is nilpotent. Note that D. J. S.Robinson and Z. Zhang [RZ] used the reduction to the subgroup C in the consideration of the monolithic case. The following problem arises in connection with the results of this chapter and Chapter 15 Question 6 < 1 >.

Describe the structure of a just non-PC-group G with Fitt(G) *

Chapter 17 Groups whose Proper Factor-Groups Have a Transitive Normality Relation

It is well known, that the relation "to be a normal subgroup" is not transitive. Therefore it is naturally to consider the groups, in which this relation is transitive. A group G is said to be a T- group, if it satisfies the following condition: (7) IfH is a normal subgroup ofG and K is a normal subgroup ofH, then K is normal subgroup ofG. In other words, a group G is a T - group if and only if every its subnormal subgroup is normal. The theory of T - groups goes back to the paper of R. Dedekind [DE 4] (clearly, a group, every subgroup of which is normal is a natural example of a T group) and has been continued in papers of E. Best and O. Taussky [BT], G. Zacher [ZG ], W. Gaschutz [GW], I. N. Abramovsky and M. I. Kargapolov [AK ], I. N. Abramovsky [A I ]. However the real progress in this area has been achieved by D. J. S. Robinson in his papers [RD 1, RD 5].Since different generalizations of r-groups have often arisen in many researches related to normality, their investigation still be very actual Continuing these researches, D. J. S. Robinson considered the soluble groups, every proper factor-groups of which is a T - group [RD 11]. Chronologically it was the third of the already investigated types of just non-^-groups. Soluble just non-r-groups play an important role in investigations related to study of the normal structure of groups and the class of such groups has been described as best as possible. In [RD 11] D. J. S. Robinson provided all details of its structure. Definitely this description requires very detail analysis and significant amount of work.. This class of groups has some peculiar properties. If just non-abelian groups were monolithic, and just infinite groups were non-monolithic, then both these cases meet combined already in the study of just

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just non-r-groups. Further, soluble T - groups are metabelian. The description of simple modules over metabelian groups in the general case is a complicated problem. Therefore it is interesting to obtain their description over some specific types of metabelian groups. The present chapter is devoted to presentation of results abut just non-r-groups from the paper [RD 11]. Since there are no too many classes of groups described so neat, we decided to keep all, even small details of their structure. For this, we will permanently apply the information on the structure of soluble T-groups given in [RD 1]. The main results about soluble T - groups will be given now. All details one can find in [RD 1]. An automorphism

maps each element to a power of itself. Let G be a soluble T- group; then it is metabelian [RD 1, Theorem 2.3.1]. Also L = [[G,G],G] is the last term of the lower central series of G, and GIL is a Dedekind group (that is the group, every subgroup of which is normal). Furthermore, CG(L) = CG([G,G]) = Fitt{G). The following three classes naturally appear in the case of non-abelian soluble T- groups (see again [RD 1]): (A) G is a periodic group; (B) C = CG([G,G]) is non-periodic (G is a non-periodic group of type I); (C) C = CG([G, G]) is periodic (G is a non-periodic group of type II). If G is a periodic soluble T - group, then L and GIL does not contain elements with the same odd prime order [ RD 1, Theorem 4.2.2] and the Sylow 2 subgroup of L is divisible. If G is a soluble T- group of type I, then C is abelian and G =< t,C >, where \G : q = 2, c' = c~l for each c e C, < t2,C2 > = < t2,C4 > [RD 1, Theorem 3.1.1]. If G is a soluble T - group of type II, then its structure is less known. However, C is abelian, [G,G] is divisible and C=[G,G]xB where B < ((G). If p e I1([G,G]) and Bp is a Sylow p - subgroup of B, then Bpp = < 1 >. If x e G, Cp is a Sylow/? - subgroup of C, c e Cp, then cx = ca, where a is an invertible/? - adic integer such that a = l(mod p"^); here ca is understood to mean c"[ where ai is an integer such that ai = a(mod \c\) [RD 1, Theorem 4.3.1]. Note also, that some other important facts about T - groups and their generalizations can be found in the survey of D. J. S. Robinson [RD 24]. We need the following technical lemma. 17.1. Lemma [RD 11]

Let G be group, H an abelian normal subgroup of G

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such that every subgroup ofCo(H) is normal in G. Assume also that for every prime p the Sylow p - subgroup ofH is either elementary abelian or not bounded. Then G is not a just non-T-group. Proof Suppose that G is a just non-r-group and let R be a non-normal subnormal subgroup of G. Then R $ Ca(H), so there exists b e R such that [b,H] * l.The element b induces a non-identity power automorphism in H, because H < CG(H). If H is not periodic, then from above hb = h~l for all h e H, and [H,,b\ = H2'. If s is the subnormal defect of R in G, then R > H1' * < 1 >, which implies that R is normal in G. Thus H is periodic and there is a prime p e Tl(H) such that a Sylow p - subgroup P of H is not centralized by b. There is a p - adic integer a such that hb = h" for each h e P [RD 1, Lemma 4.1.2]. With s as before, R > pCo-i)^ and consequently pC"-')1 = < 1 >. However, it implies that either P is elementary abelian and a = l(mod p) or P is not bounded and a = 1; in this case [P,6] = < 1 >. 17.2. Lemma [RD 11] Let G be a just non-T-group, M and U its normal subgroups. If one of the following conditions holds, then either MorU is identity. (f) M and U are periodic andU(M) n IT([/) = 0. (if) M is periodic, U is torsion-free and G/U is periodic. Proof Let H be a non-normal subnormal subgroup of G and suppose that M * < 1 >. In both cases (i) and (ii) MC\ U = < 1 >; thus (Hn M)U is normal in G. It follows that [77n M,G] < HC\ M and so Hf)M= < 1 >. Similarly Hf]U = < 1 >. If (i) is valid, then

//n(Mx LO = (/fnJW)x(i/nt/) =< 1 >, which gives H = (HM) n (HU). It follows that H is normal in G. If (ii) is valid, then H is periodic, because H = HUIU; therefore HM is periodic and (HM) C\U= , which implies that (HM) n (#£/) = # and again H is normal in G. • The study of soluble just non-r-groups is divided into some partial cases. The first case is about nilpotent just non-r-groups. 17.3. Theorem [RD 11] Let G be a nilpotent just non-T-group. Then G is a just non-abelian group with the exception of the quaternion group of order 8. Proof Suppose that G is not just non-abelian. Every nilpotent T - group is a Dedekind group, in particular, it has a finite derived subgroup (see, for example, [RD 19, 5.3.7]). Thus for this case we will use Corollary 16.4. By this Corollary G

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is torsion-free, [G, G] < £(G), £(G) is locally cyclic, and [G,G] = < d > is cyclic. Let/? be an odd prime; then the factor-group Gl < dP > is non-abelian and non-Dedekind, a contradiction. This contradiction shows that G is just non-abelian. Naturally, we must exclude the quaternion group of order 8, because it is a T - group. The next step is the consideration of soluble just non-r-groups G with no minimal normal subgroups. Denote by Z(p°°) the ring of integer p - adic numbers and by Q(px) the field of p - adic numbers. 17.4. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent and does not include minimal normal subgroups. Then Q(px) includes a subfleld F and U(Z(pco)) includes a subgroup Y, satisfying the following conditions: (i) 7 * < -1 >; (ii) ifY+ is the additive group generated by Y, then Y+ * Q + Y+ = F; (Hi) the group G is isomorphic with the natural semidirect product ofF (as an additive group) by Y. Proof Let L= [[G,G],G], then L *< 1 > and GIL is a Dedekind group. Furthermore, the lower central series of G is stabilized on L. If H is a non-identity normal subgroup of G, then GIH is a soluble T - group and hence is metabelian. Thus G" < H, and either G" = < 1 > or G" is the monolith of G. The latter is impossible. Consequently, [G, G] is abelian, and hence L is abelian too. Suppose that L is not torsion-free, let p e 11(1), P = QAi(Z,).Then P is a non-identity G - invariant subgroup of L. Since G does not include minimal normal subgroups, P must have a descending series of G - invariant subgroups P = Pi> P2> ... Pa > Pa+\ > ... Py = < 1 >, where y is a limit ordinal. Let g & G, since GIPa is a T- group, g induces a power automorphism on PIPa Let a e P\Pi, then (aPi)s = akPi for some 0 < J f c < p - l . Clearly if a > 2, then (aPa)g = akPa. It follows that CG(aPa) = CG(aP2) for all a > 2, and hence CG(PIPt) = CG(P). This means that GICG(P) is finite, that is P < FC(G). But in this case G includes minimal normal subgroups, what contradicts our hypothesis. Thus L is torsion-free. Suppose that L is not Z - irreducible. Then L has a descending series of G - invariant subgroups

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L = L\ > Ll > ... La > La+l > ... Ly - < 1 >, where y is a limit ordinal. If a > 2, then [G/La,G/La] = [G,G]/La < L/La, so GILa is a soluble T - group of type I. Put C = CG(L), then C < CG(L/La). Note that for every T - group g the equality CQ([Q,Q]) = CQ([[Q,Q],Q]) is valid [RD 1, Lemma 2.2.2]. From this we may deduce that C < CG([G,G]ILa). Since G is non-nilpotent, G * C. Choose g e CAC. Then g <£ CG{[G,G\ILO) if a is large enough. From the structure of a soluble T- group of type I we obtain that CILa is abelian and (cLa)s = c~xLa for every c e C and a > 2. It follows that C is abelian and cg = c _1 for all c e C,g e G\C. In particular, every subgroup of C is G - invariant, and Lemma 17.1 leads us to a contradiction, which proves that L is Z - irreducible. Suppose that LP * L for some prime p. Then p ±2, because Z,/L2 is divisible [RD 1, Lemma 2.4.1]. Since GIL is nilpotent, all elements of finite order of GILP form a subgroup. If GIL is not periodic, GIU is a soluble T-group of type II [RD 1, Corollary 2 of Theorem 3.1.1]. But for this case LIIf is divisible [RD 1, Theorem 4.3.1]. This contradiction shows that GIL is periodic. Put Lm = n « e z ^ " ' ^ e n L/Lm is torsion-free, the Z- irreducibility of L implies La = < 1 >• If x e D = CoiLIW), then x induces an automorphism on each LIIP" whose order is a power of p. Since GIL is periodic, we conclude that x induces an automorphism on L whose order is a power of/?, that is DIC is a p group. However GIW is a periodic soluble T - group, therefore p £ Tl(GIL) [RD 1, Theorem 4.2.2]. It follows that D = C, in particular, G/C is a cyclic group and \GIC\ divides (p - 1). Let 1 ± a e L, A = < a >G , then ,4 is finitely generated. Suppose that H{GIL) contains an odd prime q. Then GIAq is periodic, q e Il(G/L) r\U(L/Ai). This contradicts Theorem 4.2.2 from [RD 1]. Thus Tl(G/L) = {2}. Let g e CAC. Then g induces in A/A3" a power automorphism whose order is a power of 2 and divides c"1, the latter being the only power automorphism of order 2. Since c c l r D « E N ^ 3 = < 1 >> * = ~ f° every c e A, unless [^4,g] = < 1 >. Since LIA is s 1 periodic, it follows that c = c" for all c e L,g e CAC. Next Z, = L2, thus LIA2 includes a Prufer 2 - subgroup. Let PA42" be a Sylow 2' subgroup of LIA2"', n e N; then P * L and L/F is an abelian divisible 2 - group. In this case CIA2" is nilpotent, so it is a Dedekind group. But CIA2'' also includes a Prufer 2 - subgroup. This means that CIA2" is abelian, and therefore C is also abelian. We have CIA2" = PI A2" x El A2" where El A2" is a 2-group. If g e CAC, then from the structure of a soluble T-2-group [RD 1, Lemma 4.2.1] we obtain g2P e C(G/P), and hence \g2,E] < A2". Since c« = c"1 for all c e Z,,[g2,L] = < 1 >. It follows that [g2,C]. It is easy to see that g <£ CG{CIP). Using again Lemma 4.2.1 from [RD 1] we obtain that cg = c~x for each c e CIP. In other words, if c e C, then cg = c~la for some a e P, therefore, c = cgl = (c~xa)g = ((c _1 a) _1 ) a_1 = ca~2. Since L is

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186 g

_1

torsion-free, a = 1 and c = c . In particular, every subgroup of C is G - invariant, and Lemma 17.1 leads us to a contradiction. This contradiction proves that L is divisible. Using the enlarged version of Theorem 4.5 (see, [RD 22]), we obtain that G splits conjugately over L: G = L X X. Suppose that D = Cx(L) * < 1 >. Clearly, D is normal in G, so GID is a T - group. Since D does not include [G, G] and [G,G]D/D > LDID =G L, GID is a soluble T- group of type I. In this case the Zirreducibility of L implies that L = +. Moreover, L includes a G - invariant cyclic subgroup M such that CIM is abelian. For every element g e G\C, a e L, we have ag = a~l, and therefore (aM)g = (a~1)M. Since g induces a power automorphism on CIM, Tl(L/M) is the set of all primes, and since a power automorphism maps elements of the same order to the same powers, (cM)g = c~xM for every c e C. This is valid for every non-identity subgroup of M. Since the intersection of all such subgroups is identity subgroup, C is abelian and cg = c _1 for all c e C. In particular, every subgroup of C is G - invariant. Lemma 17.1 leads us to a contradiction, which shows that Cx{L) = < 1 >. Since G is metabelian, [L, [X,X]] - < 1 >. This means that Xis abelian and [G,G] = L. We can consider L as a QA' - module, and this module is simple and faithful. Choose 1 * a e Z, ; then Z, = aQX Furthermore, the QX - module L is isomorphic with QXJK where £ = Annqxia) is a maximal ideal of Q. Put A = < a >x ; then A is normal in G and GA4 is a T-group. If A: e I , 0 * n e Z, then {a^A)x = a^A for some integer /w. Therefore (-£•)*- (-f-) e ZX+ /T and QX = Q + ZX+ AT. Put <SP = {P | P is a G - invariant subgroup of L such that I/P is a non-identity p group}. Since L is not a minimal normal subgroup, we can find a prime p such that Sp * 0. It is not hard to prove that p | Sp = < 1 >. Next, let Pi,P2 ^ Sp. I f x e X ; then x induces on LIP\ and LIPj power automorphisms that can be described by p- adic integers cm and ct2. But LI(P\ f]Pi) is also a j j - group and x induces in it a power automorphism describable by ap - adic integer 03. Clearly, ai = 03 and a2 = a3,soai = a2. It follows that to each x e X there corresponds a unique p - adic integer a x such that bxP = (W) a * for all b e Z, and P e Sp. Moreover, a* = 1 if and only if x = 1, because < 1 > = [}SP = CA-(Z,). This enables us to constructs a mapping <£> : QX —* Q(pm), where Q(p°°) is the field of p - adic numbers, as follows:

( Z ^ ) * ) 0 = E^x)*,,/-, e Q. Since a^ = a*^, then O is a ring homomorphism; also it is easy verify that

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Ker. Let F = / w O ; then QX/K = F< Q(p°°),_ note that F is a subfield of Q(p°°). Put Y = X®, then Y < i/(Z(p°°)). Let G be the semidirect product of F (as an additive group) by Y. Then the mapping (ar)x —• (r and F = Q + Y+; on the other hand, F = Y+ would imply that £ is a minimal normal subgroup of G. Also y * < -1 > since otherwise F = Q and G would be a T- group. 17.5. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent, non-periodic andL = \[G,G\,G\ includes a minimal G - invariant subgroup M.IfM is torsion-free, then M is non-cyclic and G = M X X where X is a soluble T - group, CM(X) = < 1 > and M is a simple IX - module. Proof Observe first that G is monolithic with the monolith M. For otherwise G includes a non-identity normal subgroup W such that W P\M = < 1 >. Since M =G MWIW, MWIWis a minimal normal subgroup of the soluble T- group GIW, hence it is cyclic of prime order, what contrary to the assumption. Suppose that M has a complement X in G. Since Cx(M) is a normal subgroup of G and M is the monolith of G, CX{M) =< 1 >. Since X = GIM, X is a soluble T group. Further, M, as the monolith of G, could be considered as a simple IX module. Consider next the case when M has no complements. If we assume that [M, [G, G]] * < 1 >, Theorem 4.5 yields that G splits conjugate over M, contrary to the assumption. Thus [M,[G,G]] = < 1 >. If GIM is abelian, M= [G,G] because [[G,G],G] * < 1 >, using again Theorem 4.5 we obtain that G splits conjugate over M, contrary to the assumption. Hence GIM is non-abelian. Since M is torsion-free, GICQ{M) can not be periodic by Corollary 1.18. Hence GIM is not periodic. If we suppose that GIM is a soluble group of type I, then G/[G,G] would be periodic [RD 1, 3.1], and the equality [M,[G,G\] = < 1 > implies that G/CG(M) is periodic. Thus GIM is a soluble T- group of type II and [G,G]/M is periodic. Since M < £([G,G]), the set T of all elements having finite order in [G, G] is a characteristic subgroup including the derived subgroup of [G, G] (see, for example, [RD 9, Corollary to Theorem 4.12]). Since the monolith M of a group G is torsion-free, [G, G] is abelian. Clearly, M is divisible. It follows that [G,G] = Mx {/for some subgroup £/(see, for example, [RD 19, 4.1.3]). Since U is periodic, U is the periodic part of [G,G], in particular, U is normal in G. It follows, that U = < 1 > and [G, G] = M, and again obtain a contradiction, because GIM is non-abelian. Hence this case is impossible. 17.6. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent, non-periodic and L = [[G,G],G] includes a minimal G - invariant

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subgroup M. IfM is an elementary abelian p - subgroup for some prime p and [M, [G, G]] * < 1 >, then M is non-cyclic and G = M X Xwhere X is a soluble T - group, CM(X) = < 1 > and Mis a simple ZX- module. Proof Here Theorem 4.5 can be used directly to show that G splits conjugately over M. Let X be a complement to M in G and suppose that C = Cx(M) * < 1 >. Since C is normal, GIC is a soluble T - group. Moreover, M =G MCIC , in particular, \M\ = p. But in this case, [M, [G, G]] = < 1 >, what is impossible. So C = < 1 >. 17.7. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent, non-periodic and L = [[G,G],G] includes a minimal G - invariant subgroup M. IfM is an elementary abelian p - subgroup for some prime p and [M,\G,G]\ = < 1 >, then G is a group of one of the following types: (1) G = M X X where M is non-cyclic and X is a soluble T - group, CM(X) = < 1 > and M is a simple IX - module. (II)G = W\DandW = A X Qwhere (IIA)A = < a„ \apl = l,a£ + I = a„,n e N > is a Prufer p - group; (IIB) there is a monomorphism Q : Q —* U(Z(px)) such that ImQ is not periodic and a = l(mod p) for each a e ImQ; (IIC) D is a non-identity elementary abelian p - subgroup; (IID)D< C G (ai) n CG(WI < ax >); (HE) ifp = 2 and-I e ImQ, then t2 = 1 or t2 = a\ where t is apreimage of - 1 by Q; if[D,t] * < 1 >, then t2 = a\ can be dispensed with. (III)/? = landG = WX D,W = (A x< u >) X Qwhere (IIIA) A =< a„\a\ = \,a2n+x = a„,n e N > is a Prufer 2 - group, \u\ = 2; (II1B)A = [W,W\; (IHC) there is a monomorphism 6: Q —• U(Z(2m)) such that ImQ is not periodic, - 1 e Im0, and a = \(modl)for each a e ImQ; (HID) ifx e Q andxQ = \(modA), then ux = u, otherwise ux = a\u; (HIE) D is an elementary abelian 2 - subgroup; (IIIF) D < CG(ax) n CG(WI < ax >); (IIIG) t2 = aiu where t is apreimage of-I by Q; (IIIH) f (G) = < ax >. Proof We shall analyze the following cases which appear here. We start with the case when GIM is abelian. Then M = [G, G] and G splits conjugately over M by Theorem 4.5. Let X be a complement to M in G and suppose that Co = Cx(M) * < 1 >. Clearly, Co is normal in G, so that Co < C(G), because Co D [G,G] = < 1 >. Since G/Co is a soluble T- group, |MCQ/CO| = p, hence and

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\M\ = p. Therefore X/Co is finite and Co must be non-periodic. Let U be a maximal torsion-free subgroup of Co. Thus Co/t/ is periodic, consequently and GIU is periodic too. However Lemma 17.2 yields the contradiction M = < 1 > or U = < 1 >. It shows that Co = < 1 >. If M were cyclic, X would be cyclic and \X\ divides p-\, and G would be finite. Hence M is non-cyclic. So G = MX X whereXis a soluble T- group, CM(X) = < 1 > and Mis a simple ZX- module. Consider now the case when G/M is a soluble group of type I. From the structure of a soluble T - groups of type I, it is clear that G/[G, G] is periodic and [G, G]/M is not periodic and abelian. Choose a maximal torsion-free subgroup UIM of [G,G]/M. Since G/M is a T - group, U is normal in G. Also GIU is periodic. Furthermore, [U, U] < M < £(U) because [M,[G,G]] = < 1 >. For any x,y e U we have 1 = [x,y]p = [xP,y] and UP < C,(U). In particular, UP is abelian. Since UIMis torsion-free and Mis an elementary abelian, H = (Up)p is torsion-free. But clearly H is a normal subgroup of G and G/// is periodic. Lemma 17.2 again leads to a contradiction, showing that this case is impossible. Consider now the case when GIM is a soluble group of type II. Here [G, G]IM is a non-identity periodic divisible group, moreover, if C = CG([G,G]IM), then C/M is periodic and abelian. We must consider now the following two possibilities. Subcase (a): [M,C] * < 1 >. By Theorem 4.5 G splits conjugately over M. Let Xbe a complement to Min G and suppose that D = Cx(M) * < 1 >. Then GID is a soluble T - group, so it is metabelian. From the equation Mf\D = < 1 > by Remak's theorem it follows that G is metabilian. Since [G, G]IM is divisible, [G, G]p * < 1 >. Therefore GI[G, G]p is a soluble T- group in which the elements of finite order form a proper subgroup; such a group cannot be of type I. Consequently, either [G,G] = [G,G]P or G/[G,G]P is a soluble T- group of type II, in which event [G, G]/[G, G]p is divisible, an obvious absurdity. It follows that [G,G] = [G,G]P. Also [G,G] is periodic and abelian; with the aid of Lemma 17.2, we deduce that [G,G] is a divisible abelian p - subgroup. For each c e C define the mapping , we obtain that c is divisible too in the case if [G,G] * Ker(pc. But [G,G]IKerc = [G,G] for each c € C. Hence [[G,G],C] = < 1 >, in particular, [M, C] = < 1 >, which is contrary to the hypothesis. Therefore, Cx(M) = < 1 > and G = M x X where X is a soluble T - group, CM(X) = < 1 > and M is a simple IX- module. Subcase (b): [M,C] = < 1 >. Since C/M is abelian , C is nilpotent of class < 2. Lemma 17.2 implies that C is also a p ~ subgroup. For each c e C consider the mapping (pc: a —> [a,c], a e [G,G]. Again (pc is an endomorphism, Kerq>c < M. The equation [M,C] = < 1 > implies M < Ker(pc. It follows that [G,G] < (pc is divisible if it is non-identity. As above we obtain that [[G,G],C] = < 1 > and

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C = Ca(.[G,G]). In particular, [G,G] is abelian. Furthermore, [G,G] = [G,G]P, otherwise GI[G, G]p would be a soluble group of type II. Therefore [G, G] is a divisible abelian p - subgroup. Let g e G and denote by ig the automorphism induced by g in [G,G]. Now g induces in [G, G]/M a power automorphism which can be described by a p - adic number ag. Designating 9 for the power automorphism a —• aaz,a e [G,G], we can observe that i^O acts trivially on [G,G]/M. Therefore igl0-l e Hom([G,G],M) =< 0 > and ig = G. Consequently, (1)

& = a";a e [G,G],g e G.

Suppose that there exists a non-identity normal subgroup F of G such that Mf) V= < 1 >. If Cf] F = < 1 >,then [[G,G],F] < CD F = < 1 >, which is impossible. Thus C n V * < 1 >, and without loss of generality we can assume that V < C. Now G/C, and hence GIV, is non-periodic and GIM is non-abelian because M < [G,G]. Hence G/Vis also a soluble T - group of type II. It follows that CIV is abelian, and Remak's theorem implies that and C is abelian. Let g € G; then (aM)g = (aa*)M, a e [G,G], because [G,G] is a divisible abelianp subgroup. It follows that (cM)g = (c"*)M for c e C. But a» = aa« for all a e [G,G], therefore it is valid for [G,G]V/V and CIV. Hence c« = c"« for all c e C. Since C = CG([G,G]) and [G,G] is not bounded, Lemma 17.1 implies a contradiction. This means that G is a monolithic group with the monolith M. In turn, it follows that [G, G] is a Prufer/7 - subgroup. Put Z = C(C); then [G, G] < Z, and (2)

Z=[G,G]xZ)i

for some subgroup D\ (see, for example, [RD 19, 4.1.3]. Clearly D\M< C< G, so DiMis normal in G and [DUG] < [G,G] C\(DiM), which show that (3)

[DUG]<M

If d e £>i,g G G, then 1 = [rf.gp = \dP,g\, therefore D?x < C(G), in particular, D^ is normal in G. Since Mis the monolith of G, D^ = < 1 >, i.e. £>i is an elementary abelian p - subgroup. Let T/[G,G] be the periodic part of GI[G,G]; then C < T. Moreover, C * [G,G] by Lemma 17.1. The structure of a soluble T - group of type II provides the following information: {CI[G,G])P' = < 1 > for some e e N and ag = \{modpe) for all g e G [RD 1, Theorem 4.3.1]. Observe that e > 0, so that (4)

ag= \[modp).

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Now suppose that g s T . Since ag is an invertible p - adic integer satisfying (4), there are the following possibilities: either/) is odd and ag = 1 orp = 2 and ag e {1,-1}. Thus either T = C or |77q = 2, and we can write (5)

T =

where either r = 1 or a, = - 1 ; in either case t2 e C and |*2M| =1,2. Therefore *2 has order dividing 4. Let t * 1; if g e G, then / g = to where a e [G,G]. Hence (<2)g = (to)2 = f2 because a' = a'1. In particular, < t2 > is a normal subgroup of G. If f2 * 1, then M< < t2 >. Let p = 2 and [G,G] = < a„ \ 2 2 2 a\ = l,a „+l = a„,n e N >; then M= < a\ >. Clearly t e Z. If f e [G.G], then /2 = 1 or t2 = a\ because a'2 = a^1 if f * 1. Suppose t2 £ [G,G]. Since |/2| = 2or 4, it belongs to < cti > xD\. Since a\ 6 < i1 >, we can assume that f2 = a2U for some 1 * u e D\. Thus the possibilities for t2 are only \,a\,a2U. Since 77[G,G] is bounded, (6)

G/[G,G] = (TI[G,G\) x (YI[G,G\)

for some subgroup Y (see, for example, [RD 19, 4.3.9 ]. From (6) and (5) we obtain (7)

G=

Consider the case when C is abelian. Here C = Z = [G,G] x D\, and together with (7) becomes G = < t,D\,Y>. Put W = < t,Y>. Since D\ is an elementary abelian, D\ = (< t,[G,G] > C\Di) x D for some subgroup Z). Hence G = = WD. From the inclusions [G,G] < Y< W we obtain that W is normal in G. Since TC[Y= [G,G], we have Wp\D < < t,[G,G] > |~l D = < 1 >. Hence G = WD and W D £> = < 1 >. Now we consider the structure of W. First (8)

W/(WnC)= WCIC = GIC.

The map gC —• ag is an isomorphism of GIC with a non-periodic group T of p - adic integers all of which are congruent to 1 modulo/?. Now WPiC= Wf)([G,G]xDi)

= [G,G]x(fFn£>i).

Also WnDt

=< t,Y> nDi =< *,[G,G] > (iDi = < <2,[G,G] > n£>i.

If/2 € [G,G], then Wfl-Di = < 1 > by the last equation; otherwise h = aw and < t2,[G,G]> DDi = < w >, so ^ n ^ i = < « >. Hence ^ f l C = [G,G]

Just non-X-Groups

192

or WHC = [G,G] x < u > according to whether t2 e [G,G] or t2 g [G,G]. Also, W/(Wf] C) = r by (8). Suppose that t2 = « i ; if there is an element rf e £> such that td * <, then d' = da\ by (3); thus (td)2 = * 2 ai = 1, and, replacing t by ta?, we can assume that t2 = 1. In other words, we can exclude t2 = a\ unless [D,t] = < 1 >. Suppose t2 $. [G,G]. In this case p = 2 and t2 = aw where s e f l i . For all g e. G we have (a2«) g = aiu and w# = a 2 "*M because / 2 e £(G). Thus ug = u or ug = a\u according to whether ag = l(mod 4) or ag is not congruent to 1 modulo 4. Finally, D < CG(M) and D < CG(WIM). If t2 e [G, G], then D * < 1 >; for [W,W\ = [G,G] and CW{[W,W\) = WnC = [G,G]; thus W not a just non-r-group by Lemma 17.1 and D * < 1 >. Since M is the monolith of G, f(G) = < a i >. Consequently, G is of type (II) or (III) according to whether t2 £ [G,G] or t2 <£ [G,G]. It is necessary to consider the second possibility: let C is nilpotent of class exactly 2. Then C(C) = Z and C/M is abelian; thus M = [C,C] < [G,G] < Z < C. If x,y e C, then 1 = [x,y]P = [xP,y] It follows that CP < Z and C/Z is an elementary abelian p - group. Now [G, G] < Cp is immediate, and since Cp is abelian, Cp = [G, G] x F for some subgroup F (see, for example, [RD 19, 4.1.3]). Let 1 * x e F, then x = c^a for some c e C, a e M For every element g & G we have c g = c"*6 for some b e M. Hence (c^)* = (c"^b)P = (c^)"* and < c'' > is normal in G. Clearly, a =£ x; thus cp * 1 and a e < c p >. Therefore x = c^a e < cp > and < x > is normal too. Since M is the monolith, M < < x >, contradiction. Hence F = < 1 > and Cp = [G,G]. Let

C/Z = XASA < Jf^Z >

and

put

X =

< xx

|

A e A >.

Then

C = XZ = X[G,G]D\. Let y e (X[G,G]) flDil then y = *£',... * £ a where a e [G, G],«, e N and A, are distinct elements of A. Then x"k\... xn{r e Z and then p\n, for each i,I < i < r. Hence y e CP[G,G] = [G,G] and j> e [G,G] (1 £>i = < 1 >. Writing £ = X[G,G] we obtain C = £ x f l , . Suppose that xpk * 1. From the equation [G,G] = Cp we obtain ^ = ap for some a e [G,G]. This implies that (JCAGT1)'' = 1- Replacing xx by xxa~l we may assume that xpx = 1 for all A e A This implies that X f l Z = M: in fact, let y & Xf)Z; since [X,X] < M, we can write j> = x"^... x"xrra where a e Mand all A, are distinct. Thusp|«, for all / andy e M, as required. Next £CY) = M because £ W 0 Z = M Thus [X,X] = {{X) = M and X/M = C/Z. Consequently X is an extraspecial p - group. Since XC\ [G, G] = M = < a\ >, the subgroup £ is a direct product of Xand [G,G], in which the centre of X a n d < a i > are amalgamated. Return now to the element t2. If t2 £ [G,G], then p = 2 and t2 = aiit where w e Z)i. If / 2 e [G,G] and / 2 * 1, then p = 2 and f2 = a i . Suppose that [X,t] * < 1 >; then there is an index A e A such that [xx,t] * 1. Since |^AA/| = 2, [XA,*] € M and x\ — xxa\\

[X,/] = 1, write a\ = x

2

thus (txx)2 = fli^2 = 1. If,

however,

for some x eX because M = X2; then (£x)2 = 1.

Groups whose Proper Factor-Groups Have a Transitive Normality Relation

193

Therefore, if t2 e [G,G], we may assume that t2 = 1. From C = XI and (2), (7) we obtain G = < t,X,Y,Di >. Now define W = < t,X, Y >. Writing Di = (< t, [G,G] > f]Di) x D, we have G = WD and ^ is a normal subgroup of G. Suppose that w e Wp\D and, using [G,G] < Y, we will have w = t'xy where x e X and jy e Y; then jy e 7 fl K = [G, G] and t' e D, what shows that i may be assumed even. Since t1 e Z,x e Xf)Z = M and us < t, [G, G] > nD = < 1 >• Hence ff n £> = < 1 >. Consider now the structure of W. First Wf\C = Wn(ExDi) = £ x (ff fl £>i), moreover, as above, WnDi =< *,X,[G,G] > flDi =< f2,X,[G,G] > n£>! =< i2,[G,G] > nDi. If t2 € [G,G], then Wf\D\ = < 1 >; otherwise, t2 = a2u and WHD\ = < u >. Thus fFn C = E or fFn C = Ex < u > according to whether t2 e [G,G] or f2 « [G,G]. Moreover, WI{Wf\ C) = G/C and, as above, G/C = T. If/2 ^ [G,G], then one shows (as in the case C abelian) that ug = u or ug = a\u according to whether ag = 1 (morf 4) or a g is not congruent to 1 modulo 4. Since ag = \{mod p) for all g e G,[X,G] < M. Finally, C»<£>) = < 1 > and D < CG(M) n CG(WIM). Therefore again G is of type (II) or (III) according to whether?2 e [G,G]orf2 ^ [G,G]. 17.8. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent, periodic and L = \\G,G\,G\ includes a minimal G - invariant subgroup M. If [M, [G, G]] * < 1 >, then G = M\ X where M is a non-cyclic elementary abelian p - subgroup for some prime p, X is a soluble T - group, CM(X) = < 1 > and Mis a simple IX- module. We must only repeat the proof of Theorem 17.6. 17.9. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent, periodic and L = [[G,G],G] includes a minimal G - invariant subgroup M. If[M,[G,G]] = < 1 >, then G is a group of one from the following types: (I) G = M\ Xwhere Mis non-cyclic andXis a soluble T- group, CM(X) = < 1 > and Mis a simple IX- module. (II) G = P X X where P is an elementary abelian p - subgroup of order p2, p is odd prime, X a diagonal but non-scalar subgroup ofGLiip). (III) G = P\< g> where (IIIA) C(P) = < 1 >; (IIIB) P is an extra-special p - group of exponent p, p is an odd prime number;

194

Just non-X-Groups

(IIIQ if{xrf(P) | X e A} is the basis ofPIC,(P\ thenxf = x\ where 1 < n < p. (IV) G = (LxD) where (IVA) L —< a„\a\)2 = \,a^+l = a„,n e N > is a Priifer 2 - group; (IVB)D=< dxmdd1 = 1; (IVQ either t2 = 1 or t1 = a2d(that is |/| = 8); (IVD) a' = a~lforalla e L,d' = a\d. (V)G = ((Z, x D)X) < t > where (VA) L —< a„ | a\ = l,aj + 1 = a„,n e N> is a Priifer 2 -group; (VB) D=and\d\ = \or\d\ = 2; (VC) X is an extra-special 2 - group generated by elements of order 2 and O Y ) = < « i >; (VD) either t2 = 1 or t1 = a2d(ifd * 1), that is \t\ = 2 or \t\ = 8; (VE)al = a'1 for all a e L,d' = aid (if d* \\x< = x((x < a\ >)a)forall x e Xwhere a e Hom(XI < a\ >,< a\ >). Moreover, ifd * 1 we can ta&e cr = 0. Proof Since [G,G]/M is abelian, [G,G] is nilpotent. Lemma 17.2 yields that [G,G] isap- subgroup for some prime/?. In particular, G has a unique Sylov/psubgroup P. Letx 6 C/>([G,G]/M). Since CG([G,G]IM)IMis nilpotent, < JC,M> is normal in G. It follows that and [M,x] is normal in G. Assuming that [M,x] * < 1 >, we find that M = [M,x] because M is a minimal normal subgroup of G. If |x| = pr, then A/ = [M, rx] = [M,*' ] = < 1 >, a contradiction. This contradiction proves the following equation (9)

[M,CP([G,G]/M)] = < 1 > .

Subcase (A): PIM is abelian. Since [G,G] < P then P = CP([G,G]IM). Therefore from (9) we obtain (10)

[M,P]=<\>.

Also [P,P] < M and (10) shows that P is nilpotent of class at most 2. Since GIP is abelian, [[CG(P),CG(P)],CG(P)\ = < 1 > and CGCP) is nilpotent. Lemma 17.2 yields that CG(P) is &p - subgroup; hence (11)

CG(P)
Now it is necessary to consider the following two possibilities. Subcase (A I): PIM is abelian and P is abelian. Put P\ = Q.i(P) and suppose that P\ * Qi(.P). Thus i ^ is a non-identity normal subgroup, so GIP\ is a T -

Groups whose Proper Factor-Groups Have a Transitive Normality Relation

195

group. If g e G, a e PI{P\)P, then a« = aa for some a e U(Z(px)). Define now X : a —* ag,q : a —+ a", a e P, then 9 = (x_1)q is an automorphism of P acting identity on PIP\. Therefore for each a & P there is an element b e P\ such that aO = abP; hence a?Q = {abp)P = aP. Therefore (bP)6 = bP and aQP = aZ^2 = a. Hence 1 = 9P = (x~p)$p because a power automorphism commutes with every automorphism of P. By (11) CG(P) = P, so (\x\,p) = 1- Consequently ^ = gp implies that % e < q > and ^ is a power automorphism of P. By Lemma 5.2.2 of [RD 1] G is a P - group. This contradiction proves that Q2CP) = Q\(P), in other words, P is an elementary abelian p - subgroup. By our hypothesis, [G,G] = CG(M), in particular, GICaiM) is abelian. By Theorem 2.3 G/CG(M) is a locally cyclic pi- group, in particular, it is countable. Since elements of G induce power automorphisms in the elementary abelian p subgroup PIM, CG{M)/(CG(M) D CG{PIM)) is cyclic of order dividing p - 1. If g e CG(M) n CG(P/M), then g induces in P an automorphism of order 1 or p. The equation P = CG(P) implies g e P. Thus (12)

CC(W) n CG[PIM) = CGGP) = P.

It follows that GIP is countable. Then there is a family {G„/P | « e N} of finite subgroups G„ satisfying the following conditions: if n < k, then G„IP < Gt/P and GIP = Q„ e N G„/P. Since G„/F is finite, there is a finite subgroup F„ such that G„ = F„P. Moreover, we can choose these subgroups such that F„ < Ft for n < k. It follows that F = |J„ e N F„ is a subgroup and G = PF. Clearly, P C\ F„ is the normal Sylow p - subgroup of F„, therefore F„ splits conjugately over {P fl F„) : F„ = (P fl F„)X„ by Schur - Zassenhaus theorem ( see, for example, [SM , Theorem 8.10]). Let k > n, then by Dedekind modular law F„ = (F„nP)(F„r\Xk). In particular, the subgroups (F„nXk) and X„ are conjugate in F„. In other words, we can choose the subgroups X„ such that X„ < Xk for n < k. Put X = |J„eN^"- T h e n F = (PnF) \ X and hence also G = P\X. Furthermore, CX(P)"= < 1 > by (12). Consider the situation when M is the monolith of G. Assume that P * M and let a e PXM. Since GIM is a P - group, AIM = < a > M is normal in G. If M is finite, then G/CG(A) is finite pi- group. Using Maschke's theorem (see, for example, [RD 19, 8.1.2]) we obtain the decomposition A = MxB where B is a G - invariant subgroup, which is impossible. If Mis infinite, the using Theorem 4.1 we obtain again that A = Mx B for some G - invariant subgroup B, These contradictions show that P = M. Thus G is a group of type (I). Consider next the opposite situation: let G include a non-identity normal subgroup U such that Mf] U = < 1 >. If Uf] P = < 1 >, then U < CG(P) = P; therefore Uf]P* < 1 > and we can assume that U < P. Also U =G UMIM shows that every subgroup of U is G - invariant, thus we can assume that \U\ = p. Also and M =G MUIU, so that |A4| = p. Suppose now that P * MU and choose a G P\MU. Since G/Mf/ is a T - group, ^1 = < a > MU is normal in G. Since ,41

196

Just non-X-Groups

is finite, GICG{Ai) is a finite/?'- group. Using again Maschke's theorem (see, for example, [RD 19, 8.1.2]) we obtain the decomposition^ = MUx B\ where B\ is a G - invariant subgroup. Then B\MIM = GB\UIU. Let g e G, then g induces in PIM and PIU power automorphisms which both have the form c —* c", c s P, because they must agree on 5 i M M a n d B\ UIU. Hence cg = c" for all c e P; this situation is impossible by Lemma 5.2.2 of [RD 1]. Hence P = Mx U and X is isomorphic with a subgroup of GLiip) what is diagonal because X induces a power automorphism groups in M and U; this subgroup is not scalar since it does not include a group of power automorphisms in P. Clearly p is odd and G is a group of type (II). Subcase (A II): PIM is abelian and P is nilpotent of class 2. Since PIM is abelian, [P,P] = M. If g € CG(P/M), then [\g,P],P] = < 1 > by (10), therefore by Three Subgroups Lemma ( see, for example, [RD 19, 5.1.10] [g, [P,P]] = < 1 >, that is [g,M] = < 1 >. It follows that for every a e P there is an element b e M such that cfi = ab, and also a«p = ah? = a. In other words, CG{PIM)/CG{P) is a p group. But (11) yields CG(P) < P\ therefore CG(PIM) < P and (13)

CG(PIM) = P.

Next, if p = 2, a periodic group of power automorphisms of PIM has order a power of 2 and equation (13) gives P = G, i.e. G is nilpotent. Thus p is an odd prime and therefore P is a regular p - group. If we assume that G includes a normal non-identity subgroup U such that Mf] U = < 1 >, then both groups PIM and PUIU are Dedekind group. Since they are also 2'-groups, PIM and PUIU are abelian. By Remak's theorem P is abelian. This contradiction shows that M i s the monolith of G. Since P * G, we can choose an element g e G\P, then g g CG{PIM) by (13). For every element aM e P/Mvte have (aM) g = (aM) a where 1 * a e [/(Z^ 0 0 )). If a,Z> e P, then (aM)s = {aa)M,{bM)z = (ba)M; hence (14)

[a,b]g = [a a >6°] = [«,6]" 2 ,

because [ ^ M ] = < 1 >. This means that the subgroup < [a,b] > is normal in G. Since M is the monolith of G, it follows that \M\ = p. Let M = < a >. Suppose that Pp * < 1 >; then M < Pp and consequently a = bp for some b e P, because P is regular. Now & = 6"c for some c e M. In turn a* = (A*y = (6 a )^ = a". But (14) implies ag = a" ; therefore a 2 = a(modp) and a = l(modp). If we assume that [PIM)pe = < 1 > for some e e N, the congruence ap'~ = l ( m o d p e ) implies thatg induces in PIM an automorphism of order of power ofp; therefore by (13) g e P. H PIM is not bounded, a must have finite order; together with a = l(mod/>). and p > 2 this implies a = 1 (see, for example, [FL 2, Theorem 127.5]). These

Groups whose Proper Factor-Groups Have a Transitive Normality Relation

197

arguments indicate that (15)

PP=<\>.

Put Z = C,(P), clearly M < Z. Let 1 * a\ e Z; then < a\,M > is normal in G, in particular, it is finite. Using again Maschke's theorem, we come to contradiction. This contradiction proves the equation Z = M. Now PIM is elementary abelian by (15); hence P is an extra-special p - group. Choose a basis {xxM\ X e A} for PIM. Then (16)

[xx,x„] = a**">

where/is a non-degenerate alternating bilinear form. Since P = CQ(PIM), GIP is cyclic with order q dividing p - 1. Hence there is an element g such that |g| = q,G = < g > P and < g > C\P = < I >• Moreover, C(P) = < 1 >. The element g induces a power automorphism in the elementary abelian group PIM of the fornix —• x",x e P, where 1 < n < p. Thus (17)

{xxY = {xx)"a"\X e A

for certain integer rtx satisfying 0 < nx < p. Suppose that m * 0. Since / is non-degenerate, there is /i e A such that/^.p) * 0(modp). We will replace xi by a suitable element of the form x\ = (xxYix^y. It follows from (16) and (17) that {xx)s = (x^ya" where u = snx + trip + st{ n2 )_/(A,^). We want to choose s and t such that/? / 5 and/? J t but u = 0(mod/?). Consider the congruence (18)

xnx +ynn +z = O(modp)

where z = ( " )/(A,/i); notice that p K z. Since /» / rt*, we need only look for >> such that/? / (yrif, +z). If «^ = 0, anyy * 0 will do; if n^ •*• 0, we can choosey such that 1 2. Consequently (18) has a solution of the required kind. Now replace xx by xx and we arrive at the basis for PIM for which (xx)g = {xx)n for all A e A. Thus G is of type (III). Subcase (B): PIM is non-abelian. Note that if a soluble T - group is a non-abelian/> - group, then/? = 2 [RD 1, Lemma 4.2.1]. Thus P is a 2 - group. In this case L/M is a divisible abelian 2 - group. Define C = CG([G,G]/M), then C/M is nilpotent of class < 2. Since the periodic part of C/(Z(2CC)) has order 2 (see, for example, [FL 2, Theorem 127.5), then \GIC\ < 2. Let x e CO i>. By (9) the mapping ipx:a —• [a,x], a e L, is a homomorphism such that Irwpx < M and .Kerpx > M. Since L/M is divisible and M is elementary abelian, the isomorphism Imx = L. Since it is valid

198

Just non-X-Groups

for each x e Cf)P, then (19)

[L,CHP]

=< 1 >.

The inclusion L < [G, G] < C n P proves that L is abelian. Since \G/C\ < 2, C contains all elements of G with odd order. Now CIM is a Dedekind group, therefore its Sylow 2'- subgroup QIM is abelian. Assume that Q * M. If [M, Q] = < 1 >, Q is nilpotent. Since 2 is normal in G, Lemma 17.2 proves that Q is a 2 subgroup, so Q = M Hence [M, Q] * < 1 >. Suppose that Z * M, then L/M is a non-identity divisible abelian 2 - group. In this case L2 * < 1 >, therefore LIL2 is divisible, which proves that L is divisible too. For every element x e C consider again the mapping , in particular, [M,Q] = < 1 >, a contradiction. It follows that L = M and GIM is a Dedekind group, thus C = CG([G,G]/M) = G and (19) becomes [L,P] = < 1 >. Since [M,Q] * < 1 >, Theorem 4.5 implies that G splits over M : G = MX and Mf)X= < 1 >. Therefore P = P n (MX) = M ( P n X ) - Since P is non-abelian, F f l l * < 1 >. Also [M,PnX] < [L,P] = < 1 >, so PHX is normal in M r = G. Thus G / ( P n ^ 0 is a T - group and the isomorphism M = G M(P f] X)I{P D X) shows that |Af| = 2. Consequently, M < C(C?). In other words, G is nilpotent. This contradiction proves that G is a 2 - group. Equation (19) yields (20)

[L, C] = < 1 > and C = CG{L).

If we assume that GIM is a Dedekind group, then L = M and C = G, (20) implies that G is nilpotent. Also LIM is divisible whenever L * M and this, as we have already proved above, implies that L is divisible. The factor-group GIM is a soluble T - group, furthermore GIM is a non-nilpotent 2 - group. By Lemma 4.2.1 of [RD 1] G = < C,t > where f2 e C,(cM)' = c~'M,c e C,C/Mis abelian and not bounded. Together with (20) it implies that C is nilpotent of class at most 2. Define a : a —• a" 1 ,a e [ , t : a —>• as,a e L. Then r - 1 cr is identity on L/M and T_1CT - 1 e Hom(L,M) = < 0 >. Therefore r =CTand (21)

ar = a-\aeL>

which proves that f e CG (•&/)• This permits us to conclude that M < £(G). Suppose there exists a non-identity normal subgroup U such that M f] U = < 1 >. Since U =G UMIM, we can assume that \V\ = 2. Also L $ U, so GIU is not a Dedekind group and its structure is similar to that of GIM. In particular, CU/U is abelian. Since it is valid also for CIM, Remak's theorem proves that C is abelian. It follows that a' = a~x for all c e C , which is impossible by Lemma 17.1. This contradiction shows that G is a monolithic group with monolith M, in

Groups whose Proper Factor-Groups Have a Transitive Normality Relation

199

particular, L is a Priifer 2 - group and \M\ = 2. Put now Z\ = f (C). Then L < Z\ and Zi = Lx D (see, for example, [RD 19, 4.1.3]). Suppose that £> contains an element d of order 4; then d' = d~la for some ae M. Hence (rf2)' = {drxa)2 = d~2 = d2. Since d2 e Zu it follows that d2 e £(G), in particular, < t > is a normal non-identity subgroup of G. But this is impossible, because < d2 > f W = < 1 >. In other words, D is elementary abelian. This implies that DMIM < C(G/M), so that (22)

[D,G]<M.

If d e D, the mapping p] =< 1 >= [M,G]. Moreover, lm C. Now CD(G) is normal in G, so C£i(G) = < 1 >. Since Hom{GIC,M) has order 2, we must have |£>| < 2. Put D = . Let now L = < a„ | a2 = ljfl^+i = a«« e N >. If c e C, then c' = c _1 a for some a e M. Hence c'2 = (c~la)~la = c. Since G = < t,C >, it follows that t2 e C(G); in particular, the subgroup < t2 > is normal in G. Therefore, either t2 = I or M < < t2 >. Also /2 s £(G) and since <2M is centralized by /, \t2\ divides 4. Thus f 2 e < «2 > x < <^ > and the possibilities for t2 are \,a\ or a2<s? (if d * 1); indeed if cf * 1, then cf = aic/by (22) since < d > cannot be normal in G. Subcase (AII(a)): PIM is abelian, P is nilpotent of class 2 and C is abelian. Here C = Z\ = LxD, and d* 1 by Lemma 17.1. Since d' = a\d, we have (to?)2 = (f2)ai(c?2) = t2a\. Hence t2 = a\ implies (td)2 = 1.Therefore we can assume that either t2 = \ort2 = azd; \d\ = 2 or \a] = 8. Thus G is of type (IV). Subcase (A II(fi)): P/M is abelian, P is nilpotent of class 2 and C is nilpotent of class 2. Since CIM is abelian, [C,C] = M < Z\. If x,.y e C, then 1 = [x,y]2 = [x2,y]. This means that CIZ\ is elementary abelian. Choose a basis {xxZ\ | A e A}. Then (XA)2 = a^' where a e Z, and /' e {0,1}. Now a = b2 for some J e i and {xxb~1)2 = x^a -1 = d'. W r i t e r = xj.6 -1 ; then y\ = ^^' c f° r s o m e c e ^ a n d (y^)' = (y~j}c)2 = y'2. It follows that < d' > is normal in G, which can only mean that d' = 1 and y\ = 1. In short, we can assume that (23)

x\ = 1

for all X e A. Define X = < xA | A e A >. By (23) X2 = [X,X\; also C = XZU so A/= [C,q = [A;A] and M= [X,X] = X2. Suppose that ueXOZi and write M = x"k\... x\rra where a e M, «, e N, A, e A, 1 < / < r. The independence of xx,Z\ indicates that each w, is even; thus u e X2M = M. Consequently

200

(24)

Just non-X-Groups

Xf)Zi=M.

Therefore £(X) <Xn f(C) = Xfl Z, = M, and £(JQ = M Also, XIM = CIZ is an elementary abelian 2 - group. In other words, X is an extraspecial 2 - group generated by the elements of order 2. Clearly C is a direct product of X and Z\ in which Q(X) and < ai > are amalgamated. As we have already observed t2 = \,ort2 = a\, or t1 = ajd (if d * 1); in fact the second possibility can be discarded if t is chosen suitable. The argument for this has already been given in the last part of Theorem 17.7. Consider the mapping a : xM —• [x,t], x e X. Since t e CG(X/M) f] CG(M), a e Hom{XIM,M). lfd=p 1, one can assume that a = 0 and [X,t] = < 1 >. For this case if x\ = x\a\, we obtain (xxd)' = xxd while (xxd)2 = 1. Thus G is of type (V). In conclusion, observe that even if d * 1 one can still take a = 0 at the expense of losing x\ = 1; for {xiai)' = x^a2 if*l * *ANote that all types of the groups, obtained above in the Theorems 17.3 - 17.9, are just non-T-groups. But we will omit here the proof of this fact. In connection with the obtained above results the question on a structure of simple modules over soluble T- groups becomes actual. 17.10. Proposition [RD 11] Let X be a soluble T-group, F = Fitt(X), C = £(F), K a field. A simple KX - module A such that Cx{A) = < 1 > there exists if and only if there exists a simple KC - module B such that Cc(B) = < 1 >. Proof We will suppose that X is non-abelian. We recall first that F is nilpotent and F = Cx([X,X]) [RD 1, Lemma 2.2.2]. Suppose that there exists a simple KXmodule A such that Cx(A) = < 1 >. Let 0 * a e A and D = O ( a ) . Since F is nilpotent, D is subnormal in X, and so D is normal in X. For every x e X,d e D we have xd = d\x for some element d\ e D; therefore (ax)d = a(xd) = a{d\x) = {ad\)x = ax. It follows that D < Cx(A) = < 1 >. In other words, CF(A) = < 1 >. Assume now that F is non-periodic; thus X is a soluble T - group of type I. From the description of these groups given at the beginning of this chapter, we obtain that F = C is abelian and X = < x, C > where cx = c _1 for all c e C, and x1 e C. In particular, the index \X : C\ is finite. By Proposition 3.6 A includes a simple KC - submodule B. We have already proved that CQ (B) = < 1 >. Now suppose that F is periodic. Letp e Tl(C),P be a Sylowp - subgroup of C, Pi = Qi(P). Clearly C < FC(G). By Lemma 3.8 p * charK and P\ includes a subgroup J such that \P\IJ\ = p and Cored-?) = < 1 >. Obviously, J is subnormal in G and therefore normal. It follows that J = CoredJ) = < 1 >. In other words, |Pi| = p so P is a Priifer p - subgroup. It follows that C is locally cyclic qlsubgroup where q = charK. By Corollary 2.4 and Theorem 2.6 there exists a

Groups whose Proper Factor-Groups Have a Transitive Normality Relation

201

simple KC - module B such that Cc{B) = < 1 >. Conversely, let there exists a simple KC - module B such that Cc(B) = < 1 >. Put U = B ®KC KX. Let A be a KX - composition factor of U. Let H be a non-identity normal subgroup of X and suppose that Hf) C = < 1 >. Since X is metabelian, HO [X,X] implies H< C. Thus Hp[ C * < 1 >. In particular, if CG(A) * < 1 >, then and CG{A) f l C = Cc(^4) * < 1 >. Using the arguments of Chapter 3, we can obtain a contradiction. Proposition 17.10 reduces the general situation of a simple module over soluble T - group Xto the case of a simple module over abelian group. This case has been in detail considered in Chapter 2. Consider also some partial cases. 17.11. Lemma [RD 11] A finitely generated soluble just non-T-group is finite. Proof Suppose the contrary, let G be a finitely generated just non-r-group which is infinite. First of all observe that G cannot be nilpotent. Indeed, a nilpotent just non-r-group is just non-abelian by Theorem 17.3, and Theorem 11.2 implies that G must be periodic and therefore finite. Let A be a non-identity normal abelian subgroup of G. Suppose that GIA is infinite. If B be a non-identity G - invariant subgroup of A, then GIB is abelian and [G,G] < B. Hence [G,G] is a minimal normal subgroup of G and [G,G] < A. Therefore GICG([G,G]) is a finitely generated abelian group. It follows that [G, G] is a finite elementary abelian p - subgroup for some prime p (see Corollary 1.17 and Corollary 2.2). Put C = CG([G,G]), then GIC is finite by Corollary 2.2. Also [C,C] < [G,G] < £(C); thus for x,y e C we have 1 = [x,y]p = [xp,y]. This means that C < f(C), in particular, C is abelian. Since GIC is periodic, it is finite. Thus Cp is a finitely generated (see, for example, [RD 9, Theorem 1.41]). There is a number n e N such that H = (Cp)n is torsion-free. Note that GIH is finite. This contradicts Lemma 17.2. Therefore GIA is finite. Then A is finitely generated and infinite. Therefore without loss of generality we can assume that A is a free abelian. Let L = [[G,G],G], observe that I * < 1 >. I f l n ^ * < l > , then G/(L (~)A) is finite, by the first part of the proof, and, replacing A by A f] L, we may assume that A < L. Let p e U(GIL); then GIAP is a finite soluble T group; however/? e H(GIL) C\ Tl(LIAp), which is impossible [GW]. Thus Af\L = < 1 > and L = LAI A, in particular, L is finite and abelian; and hence GIL is infinite. But this situation has been shown to be impossible. 17.12. Theorem [RD 11] A finitely generated hyperabelian group G which is not a T - group has a finite factor-group which is not a T - group. Proof. Suppose that

202

Just non-X-Groups

=Ho
Ha+l... Hy = f l a < / Ha

is the series of normal subgroups of G such that GIHa is not T- group for every a G for some elements a\,... ,a„ ( see, for example, [RD 9, Lemma 1.43]). Hence Hr = Ha for some a < y, and we obtain a contradiction, which proves that GIHy is not a T- group. Zorn's lemma shows that there exists a normal subgroup A/of G which is maximal subject to GIM not being a T - group. Clearly, G/M is a just non-r-group. Since G/M is hyperabelian, it is soluble. Now we can use Lemma 17.11. Note that the case of finite soluble just non-r-groups has been considered also by C. de Vivo [VC].

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Author Index

Abramovsky, I.N. ix , 181 Alcazar, J. ix, 27 Amberg, B. 43

Gaschutz, W. ix, 181,201 de Giovanni, F. x, 27, 31, 36,39,43,67,101,111, 127, 129, 150, 165, 168, 169, 174, 177, 179 Glushkov, V.M. 143 Gonzales, M. J. ix, 27 Gorchakov, Yu. M. viii Goretsky, V.E. x Grigorchuk, R.I. ix, 143 Groves, J.R.J, ix, 155 Grunewald, F.J. vii Gupta, N. ix, 143

Baer,R. 3, 13, 113 Beidleman,, J.C. viii Best, E. ix, 181 Brenner, J.L. 147 Bride, I.M. ix Brookes, C.J.B. 132 Cartwright, M. ix Charin,V.S. 82,88 Chernikov, S.N. viii Clifford, A.H. 31 Cossey, J. 162 Curtis, C.W. 31,49,58, 124, 144, 151, 152, 171

Hall, P. 3, 5, 7, 12, 134, 145, 146 Hartley, B. 19, 29, 39, 145 Hawkes,T. 34,35,36 Hirsch, K. vii, 131 Huppert,B. 68

Dedekind, R. 181 Dixon, J.D. viii Doerk,K. 34,35,36 Duan, Z.Y. 39

Kaplansky, I. 41, 53, 57, 59, 78,81,103, Karbe, M. 66,82,88 Kargapolov, M. vii, 86, 181 Karpilovsky, G. 4, 7, 18, 21,23,41,47,48,51,71, 86, 101, 102, 104

Fedorov, Yu.G. 143 Franciosi, S. x, 27, 31, 36, 39, 43,67, 101, 111, 127, 129, 150, 165, 168, 169, 174, 177, 179 Fuchs,L. xii, 24, 111 , 113, 174, 178, 196, 197

221

222

Kazarin, L.S. xi, 39 Kegel, O.H. 133,150, 162, 166 Kovacs, L.G. 39, 166 Kurdachenko, L.A. ix, x, xi, 3,12,14,27,31,33,38,39, 67, 69, 71, 82, 83, 84, 87, 88, 101,110,111,127,128,129, 138, 139, 140, 141, 149, 151, 152, 168, 170, 171, 172,173, 174, 175, 178, 179 Lang, S. 156 Lennox, J.C. viii Macdonald, I.D. viii Mal'tsev, A.I. vii, 27, 38, 85 Mann, A. viii Maschke, H 27,39,124,144, 151, 152, 171, 195, 196 Matlis, E. 4 Maznichenko, V.A. 39 McCarthy, D. ix, 143 McDougall, D. 19 McLain, D.H. 144 Menegazzo, F. ix Merzlyakov, Yu.I. 86 Murach, M.M. 71 Narkiewicz, W. 10, 79, 125 Neumann, H. vii Neumann, B.H. viii Neumann, P. viii Newman, M.F. ix, 39,115, 116 Northcott, D.G. 64

Otal, J. ix,x,27,31,33,38, 69,71, 110,111,138, 139, 140, 141, 168, 170, 171, 172, 173, 174, 175, 178,179

Passman, D.S. 5 , 6 , 9 , 1 1 , 45,55,56,58,59,81,92,99, 104, 132 Pena, M. ix, 27 Petrenko, B.V. 3, 12, 14, 39 Pickel, P.F. vii, 185 Polovicky, Ya.D. ix, 27, 32, 33, 36, 166, 168, 169, 172, 173, 174 Pylaev, V.V. ix Reiner, I. 31,49,58, 124, 144, 151, 152, 171 Remak , R. 64, 67,109, 124, 129, 136, 138, 157, 169, 189, 190, 196, 198 Remeslennikov, V.N. vii Romanovsky, N.S. vii Robinson, D.J.S. vii, viii, ix,x,xi, 11,22,25,27,32, 38,43,45,89,91,92,93, 94, 95, 96, 97, 98, 99, 100, 109, 111, 112, 113, 115, 124,131, 132, 133,134, 135,136, 138, 149, 151, 153, 155, 156, 158, 159, 160, 161, 162, 163, 165, 166, 168, 174, 177, 176, 180,181,182,183,184, 185, 187, 188, 190, 191, 192, 193, 195, 196, 197, 198, 199,200,201,202

Author Index

Roseblade, J.E. 92, 94, 99, 156 Salse, L. xii Samuel, P. 14 du Sauton, M.P.F. viii Schur, I. 17,111,149, 168, 176 Segal, D. vii, viii, 92, 131,133, 134 Shalev, A. viii Sharp, R. 84 Sharped, W. 41,50,106 Sidki, S. ix, 143 Smith, H. viii Subbotin, I.Ya. ix, x, 3, 12, 14,27,39,111,127,128,129, 138, 139, 140, 141, 149, 150, 151, 152 Suzuki, M. 195 Taussky, O. 181 Tomkinson, M.J. viii, ix, 27, 36,39,73,120,177 Tushev, A.V. 82,83,84,87

223

Vamos, P. 41, 50, 106 Vaughan - Lee, M.R. viii deVivo, C. 202 Wehrfritz,B.A.F. 66,71, 72,85,93,96, 133,152,157, 166 Wiegold, J. viii Wilson, J.S. ix, xi,45, 68, 91,92,93,94,95,96,97,98, 143, 148, 155, 156, 158, 159, 160, 161, 163, 163 Zacher, G. ix, 181 Zaitsev, D.I. viii, 3, 12, 13, 29, 39, 50, 82, 83, 84, 87, 140, 169 Zariski, O 14 Zhang Z. x, 22, 89, 163, 166, 176, 177, 180

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Subject Index

Abnormal subgroup 135 Complement (to submodule) 39 Complemented submodule 39 Condition Max - G 137 Condition Min - G 137 DedekindZ-domain 133 Dedekind Zo -domain 61 Dedekind Z\ -domain 77 Derivation 97 Derivation irreducible 99 FC -center of group 28 FC -hypercenter of group 28 Formation of groups 28 Group splits over its normal subgroup 42 Group nearly splits over its normal subgroup 163 Group conjugately splits over its normal subgroup 42 Groups CC -groups 28 FC -groups 28 FC -hypercentral 28 generalized minimax 137

225

226

Subject Index

hypercentral 25 hyperfinite 33 just infinite 143 just non-abelian 115 just non-CC-groups 165 just non-(central-by-finite) 165 just non-Chernikov 148 just non-FC-groups 165 just non-(finite-by-abelian) 165 just non-hypercentral 121 just non-hyperfinite 149 j ust non-nilpotent 121 j ust non-(polycyclic-by-finite) 155 just non-r-groups 181 just non-A'-groups 107 minimax abelian 79 minimax 136 monolithic 109 non-monolithic 109 of finite 0-rank 11 T-groups 181 A"C-groups 28 AfC-hypercentral groups 228 Hyperbolic plane 118 /-component of module 4 Isotropic subspace 118 Modules co-(Iayer-finite) 101 /-module 4 / -pure submodule 101 just infinite 45 j ust non-hypercentral (j ust non-/?G-hypercentral) 121 just non-nilpotent (just non-/?G-nilpotent) 121 minimax 78 Priifer P-module 77 P-basic submodule 55 p-periodic 4 pure submodule... 53 rationally irreducible 42 /^-irreducible 42

Subject Index

^-periodic 4 7?-torsion-free 4 i?G-monolithic module 122 i?G-non-monolithic module 123 TfG-hypercentral 84 #G-nilpotent 84 Simple 1 x-pure submodule 53 Z-irreducible 42 Monolith of a group 104 Nearly complemented subgroup 163 0-rank of a group (torsion-free rank) 11 Orthogonality 118 Orthogonal complement 118 Orthogonal direct sum 118 Pronormal subgroup 135 Plinth 92 Quasi-socle 32 i?-free subset 6 i?-periodic part of a module 4 i?-rank of a module 6 .RG-monolith of a module 122 Symplectic space 117 Upper FC-hypercenter of a group 28 Upper FC-central series of a group 28 Upper 7?C-central series of a module 84 Upper /?C-hypercenter of a module 84 Upper A'C-hypercenter of a group 28 Upper A"C-central series of a group 28 A'-conjugacy classes 28 A"C-center of a group 28 AC-hypercenter of a group 28 A'-residual of a group 42

GROUPS WITH PRESCRIBED QUOTIENT GROUPS AND ASSOCIATED MODULE THEORY The influence of different gomomorphic images on the structure of a group is one of the most important and natural problems of group theory. The problem of describing a group with all its gomomorphic images known, i.e. reconstructing the whole thing using its reflections, seems especially natural and promising. This theme has a history that is almost a half-century long. The authors of this book present well-established results as well as newer, contemporary achievements in this area from the common integral point of view. This view is based on the implementation of module theory for solving group problems. Evidently, this approach requires investigation of some specific types of modules: infinite simple modules and just infinite modules (note that every infinite noetherian module has either an infinite simple factor-module or a just infinite factor-module). This book will therefore be useful for group theorists as well as ring and module theorists. Also, the level, style, and presentation make the book easily accessible to graduate students.

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