Injectors of finite groups

~-INJECTORS

OF FINITE GROUPS

Ya. L. Fel'dman

UDC 519.44

All the groups under consideration are finite. Let ~ be a Fitting class. We will say that a group G has the C~-property if it contains ~-injectors and any two ~-injectors in G are conjugate. The basis theorem of the theory of Fitting classes (see [i]) ensures that each solvable group has the C~-property. The result of Gaseh~tz, dual to it, from the theory of formation has been extended independently in [2] and [3], where it is established that a group G has ~-projectors and any two ~-projectors in G are conjugate if the ,~-coresidual GS?, where ~O is a local formation, is solvable. Naturally the following question is posed for the Fitting class 3: also: If the quotient group G/G~ modulo the J-radical is solvable, then does the group G itself have the C~-property? For a locally defined Fitting class ~ an affirmative answer can be obtained from the fundamental theorem of [4]. However, as observed in [5], not even each subgroup-closed Fitting class is locally defined. In this article we will study the above question for the Fitting class ~., equal to the product ~ of the Fitting classes .~ and ~), where .~) = {O : G/G.~. C ~)} In particular, it is established that if a group G is such that the quotient group GIGs. is solvable and ~ ) ~ ~J~, then it has the C ~)-property. '~ The structure of the ~)-injectors is indicated. Here, and in what follows, 9~ denotes the class of nilpotent groups and J~ denotes the class of nilpotent ~-groups. See [5] for all the definitions and notation. LEMMA i.

Let G be a group, F(G)_AH_AG, and the quotient group G/H be solvable.

Then

CG(n)~_H. Proof. Let us set r = r and F(G) =F. Then r176 The lemma of [6] contains the following result: If S/~ is the product of all minimal normal subgroups of the group G/~, then C G ( S ) ~ S . Therefore, it is sufficient to prove that H ~ S . Let N/~ be a minimal normal subgroup of the group G/~. Let us assume that N~_H. Then N ~ H = O and N / ~ N H / H . But NH/H is a minimal normal subgroup of the solvable group G/H and is therefore Abelian~ Therefore N/~ is also Abelian; whence, by a well-known theorem of Gasch'utz [7], the subgroup N is nilpotent. But then N - - F ~ H , which is a contradiction. Thus, N ~ H and, since the subgroup N/~ has been chosen arbitrarily, it follows that S--H. The lemma is proved. COROLLARY i. Let G be a group, ~ be a Fitting class,and ~ J ~ , the quotient group G/G~ is solvable, then C~;(G~)~_G~.

where

~(G/G~).

If

Indeed, let F ( G ) = F = F ~ F ~ ' , where F~ is a Hall ~-subgroup of F. It is clear that F ~ G~ and, since F/F N G~-~FG?JG~92~, it follows that F.~,~F ,Q G~__G3. Thus, F~_G~ and by Lemma 1 we have C~(G~) _ G~. LEMMA 2 If the group

Let G be a group, ~ and ~ be Fitting classes, and ~ ~ J ~ , where ~ = solvable and G I~A ~ V ~ G , then

z~(G/GE~),

G/GE~ is

1) g~ -~O~; 2) the subgroup V is ~ - m a x i m a l

in G if and only if

VfG?s is ~-maximal in G/G~. H / K = (O/l()~ and (G/If)/(G/IO~ ~G/H

Proof. i) Let us set H = Oi~)and K = G l= H~. Since is a so-~-~vable group, by virtue of Corollary 1 it follows that CG/K~(H/K)~H/K. Sin~e IVy, H]-----V~H=H~=K, it follows that V~_Co(H/IO~_H. Consequently, V~__H~. The reverse inclusion is obvious. 2) Let us assume that V is an ~ - m a x i m a l subgroup of G. By the above statement we have V ~ O ~ and therefore V/G~ ~. If V/G?~~__W/G~~__G/~ and W / ~ , then W~ = 0~ implies that W ~ . Consequently, W = V and, therefore, V/G~ is ~-maximal in G/G~. Ukrainian Hydroengineering Project, Ministry of Housing and Communal Services of the Ukrainian SSR. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 32, No. 2, pp. 265-267, March-April, 1980. Original article submitted August 9, 1978.

0041-5995/80/3202-0173507.50

9 1980 Plenum Publishing Corporation

173

Conversely, let V/G~ be ~-maximal in G/G~. The equality V~ = G~ implies that F C ~ . If V ~ W ~ G and W E ~ , then W~ = 02 and W/G~6~. Hence W =V, and the lemma is proved. THEOREM i. Let G be a group and ~ and ~ be Fitting classes, ~___~, where v = ~(G/G~), and let the group G/G~ be solvable. If the group O/O 2 has the C~-property, then the group G has the C~-propert~; moreover, the subgroup V of G is an ~ - i n j e c t o r of G if and only if V/G~ is a~-injector of the group G/G~. Proof. By the condition the group G/G~ has ~-injectors; let V/G~ be one of them. We then show that the subgroup V is anX~-injector of the group G. Let N be an arbitrary subnormal subgroup of G. Then N~ = N N G~ and N/N~ ~NG~/G~. The subgroup V N N/N~ of N/N~ is isomorphic to the subgroup (VNN)G~/G~, which is obviously a ~-injector of the group NG~/G~. Consequently, V N N / N ~ is a~-injector and is, therefore, a ~ - m a x i m a l subgroup of the group N/N~. Since N ~ = G ~ N N ~ V N N and N / N ~ N G ~ / G ~ is a solvable group, it follows by Lemma 2 that V n N is an ~ - m a x i m a l subgroup in N. By the same token it is proved that the subgroup V is an ~ - i n j e c t o r of G. Conversely, let us assume that the subgroup V is a n ~ - i n j e c t o r of G. We show that is a ~-injector of the group G/G~. Let N/G~ be an arbitrary subnormal subgroup of G/G~. Then we again have N ~ = N N G ~ N N V, and by Lemma 2 we have (NNV)~ = N ~ = G~. Since N ~V, being an ~-injector in N, is ~ - m a x i m a l in N, it follows that N n V/G~ is ~-maximal in N/G~. Thus, V/G~ is a~-injector of the group G/G~.

V/G~

Let V~ and V~ be two different ~ - i n j e c t o r s of G. Then the subgroups V~/G~ and V~/G~, being~-injectors of the group G/G~, which has the C~-property, are conjugate in it. Consequently, the subgroups V~ and V~ are conjugate in the group G. The theorem is proved. We get the following corollary from the above theorem and a result of Fischer, Gasch~tz, and Hartley (see [i]). COROLLARY 2. Let G be a group and ~ and ~ be Fitting classes such that ~_____~. If the quotient group G/G~ is solvable, then G has the C~-property; moreover, a subgroup V is an ~-injector in G if and only if V/G~ is a~-injector in G/G~. Let us observe that for a solvable group G this structure of ~ - i n j e c t o r s has been obtained in [5]. LITERATURE CITED l.

2. 3. 4. 5. 6.

7.

174

B. Fischer, W. Gasch~tz, and B. Hartley, "Injectoren endlicher aufl~sbarer Gruppen," Math. Z., 102, No. 5, 337-339 (1967). E. F. Shmigirev, "On certain questions of the theory of formations," in: Finite Groups [in Russian], Nauka i Tekhnika, Minsk (1975), pp. 211-225. P. Schmid, "Lokale Formationen endlicher Gruppen," Math. Z., 137, No. i, 31-48 (1974). F. R. Schnackenberg, "Injectorsof finite groups," J. Algebra, 30, Nos. 1-3, 548-558 (1974). B. Hartley, "On Fischer's dualization of formation theory," Proc. London Math. Soc., 19, Part 2, 193-207 (1969). L. A. Shemetkov, "Outer saturation of homogeneous formations," in: All-Union Algebraic Symposium; Abstracts of Report [in Russian], Part i, Nauka i Tekhnika, Gomel (1975), p. 79. W. GaschHtz, "Uber die r endlicher Gruppen," Math. Z., 58, 160-170 (1953).