On the Shemetkov Schmid subgroup and related subgroups of finite groups

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1 Известия Гомельского государственного университета имени Ф. Скорины, 3 (84), 2014 УДК On the Shemetkov Schmid subgroup and related subgroups of finite groups A.F. VASIL'EV, V.I. MURASHKA В работе установлены свойства подгруппы Шеметкова Шмида F ( и ) связанных с ней обобщенных подгрупп Фиттинга конечных групп. Мы называем подгруппу H R-субнормальной в группе, если H субнормальна в H, R. Изучены группы с заданными системами R-субнормальных подгрупп, если R { F ( ), F, F ( )}. Найдены новые характеризации нильпотентных и сверхразрешимых групп. Ключевые слова: конечная группа, подгруппа Фитинга, подгруппа Шеметкова Шмида, квазинильпотентный радикал, нильпотентная группа, сверхразрешимая группа. In this paper the properties of the Shemetkov-Schmidt subgroup as well as generalized Fitting subgroups related with it have been determined. We call a subgroup H R-subnormal in a group, if H is subnormal in H, R. Finite groups with given systems of R-subnormal subgroups have been studied for R { F ( ), F, F ( )}. New characterizations of nilpotent and supersolvable groups have been obtained. Keywords: finite group, the Fitting subgroup, the Shemetkov Schmid subgroup, R-subnormal subgroup, nilpotent group, supersoluble group. 1 Introduction. All the considered groups are finite. In 1938 H. Fitting [1] showed that a product of two normal nilpotent subgroups is again nilpotent subgroup. It means that in every group there is the unique maximal normal nilpotent subgroup F ( ) called the Fitting subgroup. This subgroup has a great influence on the structure of a solvable group. For example Ramadan [2] proved the following result. Theorem 1.1. Let be a soluble group. If all maximal subgroups of Sylow subgroups of F ( ) are normal in then is supersoluble. Analyzing proofs of such kind's theorems in solvable case one can note that the following properties of the Fitting subgroup F ( ) are often used: (1) C ( F ( )) F ( ); (2) Φ F ( ) and F ( / Φ ) = F ( )/ Φ( ); (3) F/ Φ Soc( / Φ( )). But only (2) and (3) are held for the Fitting subgroup of an arbitrary group. Note that there are many groups with F ( ) = 1. That is why there were attempts to generalize the Fitting subgroup. In 1970 H. Bender [3] introduced the quasinilpotent radical F ( ). It can be defined by the formula F / F ( ) = SocC ( ( F ( )) F ( )/ F ( ) and can be viewed as a generalization of the Fitting subgroup. For F ( ) the statements like (1) and (3) are held. This subgroup proved useful in the classification of finite simple groups. Also F ( ) was used by many authors in the study of nonsimple groups. In 1985 P. Förster [4], [5] showed that there is the unique characteristic subgroup F ( ) ( F in Förster notation) in every group which satisfies the statements like (1)-(3). Firstly subgroup with this properties was mentioned by P. Schmid [6] in It was defined in explicit form by L. Shemetkov in 1978 [7, p.79]. P. Schmid and L. Shemetkov used this subgroup in the study of stable groups of automorphisms for groups. Definition 1.2. The Shemetkov-Schmid subgroup F ( ) of group is defined as follows: (1) F Φ( );

2 24 A.F. Vasil'ev, V.I. Murashka (2) F / Φ = Soc( / Φ( )). Proposition 1.3 [8], [9]. F F ( ) for any group. The following example shows that in general case F ( ) F ( ). Example 1.4. Let A 5 be the alternating group on 5 letters and K = F 3. According to [10] there is a faithful irreducible Frattini K-module A of dimension 4. According to known aschütz theorem [11], there exists a Frattini extension A R such that A is -isomorphic Φ ( R) and R/ Φ ( R). From the properties of module A it follows that FR ( ) =R and F ( R ) = Φ( R ). In this paper we continue to investigate the properties of the Shemetkov Schmid subgroup and related subgroups. There is an overview of some applications of considered subgroups. 2 Preliminaries. We use standard notation and terminology, which can be found in [12], [13] if necessary. Recall that for a group, Φ is the Frattini subgroup of ; is the intersection of all maximal abnormal subgroups of and ; Z ( ) is the center of ; Z is the hypercenter of ; Soc( ) is the socle of ; F is the F-radical of for a N0 -closed class F with 1; F is the F-residual of for a formation F; N is the class of all nilpotent groups, N is the class of all quasinilpotent groups; F p is a field composed by p elements. A class of group F is said to be N0 -closed if AB, and AB F, imply AB F. A class of group F is said to be Sn -closed if A and F imply A F. Lemma 2.1 [12, p. 127]. Let be a group. Then C F F ( ( )) ( ). Lemma 2.2 [7, p. 95], [14]. Let be a group. Then ( / Φ ) = / Φ = Z ( / Φ ( )) = = Z ( / Φ( )) and is nilpotent. 3 Properties of F ( ) and related subgroups. It is well known that FF ( ( )) = F ( ) and F ( F ) = F ( ). In [4] P. Forster showed that there is a group such that FF ( ( )) < F ( ). He shows that there is a nonabelian simple group E which has F p E -module V such that R = Rad( V ) is faithful irreducible F p E -module and V / R is irreducible trivial F p E -module. Let H be the semidirect product VλE. Then H = RE is a primitive group and H : H = p. There is F q H -module W with C ( ) = H W H, where q p. Let = WλE. Then Φ =Φ ( H) = R and Soc( / R) = W ER / R. So F = W ER and Φ ( F ( )) = 1. It means that FF ( ( )) = Soc( F ( )) = = RW < F ( ). This example led us to the following definition. Definition 3.1. Let be a finite group. For any nonnegative integer n define the subgroup n F by: F 0 n n 1 = and F = FF ( ) for n > 0. i i 1 i 1 It is clear that F F = = F ( F ) for some i. So we can define the subgroup F 0 1 as the minimal subgroup in the series = F F ( )... Now F = F ( F ( )). Proposition 3.2. Let n be a natural number, N and H be normal subgroups of a group. Then: n 1 (1) If N Φ( F ) then ( / ) = F N F / N ; (2) F F ; n 1 (3) If Φ ( F ) = 1 then F F ( ) = ( ); (4) C ( F) F ( ); (5) ( ) F N F ( ); (6) ( ) / F N N F ( / N ); (7) If = N H then ( ) = ( ) F F H F ( N ).

3 On the Shemetkov Schmid subgroup and related subgroups of finite groups 25 Proof. (1) When n = 1 it directly follows from the definition of F ( ) and Φ ( / N) =Φ/ N. By induction by n we obtain this statement. (2) The proof was proposed by L. Shemetkov to the authors in case if n = 1. Let a group be the minimal order counterexample for (2). If Φ( ) 1 then for / Φ the statement is true. From F / Φ F ( / Φ ) and F ( / Φ ) = F ( )/ Φ we have that F F ( ). It is a contradiction with the choice of. Let Φ = 1. Now F = Soc( ). By X [12] F = EF ( ) ( ). Note Φ ( E ( )) = 1. Since 13.7.X [12] E ( )/ ZE ( ( )) is the direct product of simple nonabelian groups, ZE ( ( )) = FE ( ( )). From the theorem 10.6.A [15] we conclude that E ( ) = HZE ( ( )), where H is the complement to ZE ( ( ) in E ( ). Now H is the direct product of simple nonabelian groups. Since Hchar E ( ), we have H. Note H Soc( ). Since ZE ( ( )) F ( ) F ( ) and H Soc( ), it follows that E ( ) F ( ). Now F = EF ( ) ( ) F ( ). It is a contradiction with the choice of. Assume that n F F for n 1. It means that F ( F ) = F ( ). By induction ( ) + ( ). n 1 (3) If Φ ( F ) = 1 then F is the socle of 1 F F From n F F it follows that F F n F 1 and hence n F ( ) is quasinilpotent. ( ) = ( ). ( ( )) ( ( )). (4) From n F F it follows that C F C F Since C F F ( ( )) ( ) by lemma 2.1, we see that C ( F) F ( ). (5) Since Φ( N) Φ( ), we see that F ( / Φ ( N)) = F ( ) / Φ( N ). Note that F ( )/ Φ( N) is quasinilpotent. Hence FN ( )/ Φ( N) F( / Φ( N)) F ( / Φ ( N)) = F ( )/ Φ( N ). Thus FN ( ) F ( ). By induction ( ) F N F ( ). (6) Note that FN ( ) NΦ( N ) N FN ( ) Φ( N ) F ( ) F ( ) Φ( N ). From Φ F ( ) Φ( N ) it follows that FN ( ) / N / Φ ( N ) / N is quasinilpotent. Since Φ( N ) / N Φ( / N ), we see that FN ( ) / N F ( / N ). Assume that n n F N / N F ( / N) for some n 1. Now ( / ) = ( ( / )) ( ( ) / ) ( ( ) ) / ( ( )) / = F N FF N FF N N FF NN N FF N N F ( N ) / N by the previous step and (5). (7) Assume that the statement is false for n = 1. Let a group be a counterexample of minimal order. Assume that Φ = 1. Then / Φ = HΦ/ Φ NΦ/ Φ( ). Note that HΦ/ Φ H / H Φ = H / H ( Φ ( H) Φ ( N)) = H / Φ( H ). By analogy NΦ/ Φ N / Φ( N ). So F ( / Φ) FH ( / Φ ( H)) FN ( / Φ( N )). From F ( / Φ ) = F ( )/ Φ, FN ( / Φ ( N)) = FN ( )/ Φ( N) and FH ( / Φ ( H)) = FH ( )/ Φ( H ) it follows that F ( )/ Φ FH ( )/ Φ ( H) FN ( )/ Φ( N ). Now F ( ) FH ( ) FN ( ) = Φ Φ( H) Φ( N) From Φ =Φ ( N) Φ ( H) it follows that F ( ) = FN ( ) FH ( ). From (5) it follows that FN ( ) F ( ) and FH ( ) F ( ). Thus F ( ) = FH ( ) FN ( ), a contradiction. Now Φ ( ) = 1. So F = F( ), F ( N) = F ( N) and F ( H ) = F ( H ). It is well known that F ( H N) = F ( H) F ( N ), the final contradiction. By induction we have that ( ) = ( ) F F H F ( N ).

4 26 A.F. Vasil'ev, V.I. Murashka From proposition 3.2 properties of F ( ) follow. Corollary 3.3. Let N and H be normal subgroups of a group. Then: (1) [9] If N Φ then F ( / N) = F ( )/ N ; (2) [8], [9] F F ( ); (3) [4] If Φ = 1 then F = F( ); (4) [6], [7] C( ( )) F F ( ); (5) [4] FN ( ) F ( ); (6) [4] FN ( ) / N F ( / N ); (7) If = N H then F ( ) = FH ( ) FN ( ). Also we obtain new properties of F ( ). Corollary 3.4. Let N and H be normal subgroups of a group. Then (1) F / Φ( F ) is quasinilpotent; (2) F F F ( ); (3) If Φ ( F ) = 1 then F = F ( ); (4) C( F ) F ( ); (5) F ( N) F ( ); (6) F N / N F ( / N ); (7) If = N H then F = F ( H) F ( N ). In [4] Förster introduced a class ˆN = E Φ N = ( F ( ) = ) and showed that ˆN is N 0 -closed Shunck class that is neither formation nor sn -closed. Note that ˆN = ( F = ). Proposition 3.5. Let be a group. Then ˆ = F, i.e. F is the maximal among N normal subgroups N of such that N / Φ ( N) is quasinilpotent. Proof. From ˆ / Φ( ˆ ) N and Φ( N N ˆ ) Φ it follows that N F ( ). By induction N ˆ = F ( ˆ ) F. By proposition 3.2 and the definition of ˆN we obtain N N = F. N Problem 3.6. Let F be an N0 -closed class of groups and 1 F. Then there is the maximal normal F-subgroup F in any group. In the context of our work the following general problem appears: to describe all N0 -closed classes (formations, Fitting classes, Shunck classes) F with 1 for which one of the following statements holds: (1) F ( ) F F for any group ; (2) F F F ( ) for any group ; (3) F ( ) F F ( ) for any group. Theorem 3.7. Let F be a N 0 -closed formation. Then: (1) If F is a saturated formation and F ( ) F F ( ) for any group then F = N. (2) If F F F ( ) for any group then F = N. Proof. Let prove (1). From F() F it follows that N F. Assume that the set F\N is not empty and we choose a minimal order group from it. Since F and N are both saturated formations, from minimality of we may assume that Φ() = 1 and there is only one minimal normal subgroup of. From F F ( ) it follows that = Soc() is nonabelian simple group. From [10] it follows that for prime p dividing there exist faithful irreducible F p -module A admitting a group extension A E with A Φ(). Since F is a saturated formation so E F and A/1 is chief factor of E. According to [15, p. 335] we see that H=A λ (E/A) F. Note that FH ( ) =A, a contradiction. Thus N = F. Let prove (2). From F () F it follows that N F. Assume that the set F\N is not empty and is a group a minimal order from it. Since F and N are both formations, from minimality of

5 On the Shemetkov Schmid subgroup and related subgroups of finite groups 27 we may assume that there is only one minimal normal subgroup N of. If Φ() = 1 then = Soc() N, a contradiction. So N Φ(). Now N is a normal elementary abelian p-subgroup. By our assumption /N N. Assume that C (N) =. Now acts as inner automorphisms on N/1 and on every chief factor of /N. By definition of quasinilpotent groups N, a contradiction. Hence C (N). Note that N is the unique minimal subgroup of H = N λ (/C (N)) F by [15, p. 335] and Φ(H) = 1. So FH ( ) =N and H F = H, a contradiction. Thus N = F. Let consider another direction of generalization of the Fitting subgroup. A subgroup functor is called m -functor if contains and some maximal subgroups of for every group. Recall [16, p. 198] that Φ is the intersection of all subgroups from ( ). Definition 3.8. Let be m -functor. For every group subgroup F is defined as follows: 1) Φ F ( ); 2) F / Φ = Soc( / Φ ( )). If is the set of all maximal subgroups of for any group then we obtain the definition of F ( ). If is the set of all maximal abnormal subgroups and for any group then Φ = ( ). Subgroup F = F was introduced by M. Selkin and R. Borodich [17]. Proposition 3.9. Let be a group. Then F ( ) and F ( / ) = F ( )/ ( ). Proof. From lemma 2.2 it follows that F ( ). Let a group be a counterexample of minimal order to the second statement of proposition. Assume that Φ( ) 1. By inductive hypothesis F (( / Φ) / ( / Φ )) = F ( / Φ) / ( / Φ( )) Now ( / Φ ) = / Φ( ) by lemma 2.2 and F ( / Φ ) = F ( )/ Φ by corollary 3.3 (1). F (( / Φ/ ( / Φ )) = F ( / Φ/ / Φ) F ( / ) F ( / Φ)/ ( / Φ ) = F ( )/ Φ/ / Φ F ( )/ Hence F ( )/ = F ( / ( )). By corollary 3.3 (6) it follows that F ( )/ F ( / ). Thus F ( )/ = F ( / ( )), a contradiction. Now Φ ( ) = 1. It means that = Z Soc( ). So Z ( ) is complemented in. Let M be a complement of Z ( ) in. Hence M and = M Z ( ) = M ( ). Note that M / ( ). From corollary 3.3 (7) it follows that F ( ) = FM ( ) F ( ) = FM ( ) ( ). Thus F ( / ) FM ( ) F ( )/ ( ). From corollary 3.3 (6) we obtain the final contradiction. Corollary Let be a group. Then F = F ( ). Proposition Let be a group. Then F F ( ). In particular C ( ( )) F F( ). Proof. From corollary 3.3 it follows that if N and Φ N then FN ( ) / N is quasinilpotent. Also note that Φ( / Φ ( )) = 1. By corollary 3.3 F ( / Φ ) = Soc( / Φ( )). Hence F ( ) Φ ( )/ Φ ( ) Soc ( / Φ ( )). Thus F ( ) F ( ). The second statement follows from (4) of corollary Applications of Shemetkov-Schmid subgroup and related subgroups. In [8] A. Vasil'ev and etc. proved the following theorem. Theorem 4.1. The intersection of all maximal subgroups M of a group such that MF = is the Frattini subgroup Φ ( ) of. Another direction of applications of generalizations of the Fitting subgroup is connected with the following concept. Recall [16], that a subgroup functor is a function which assigns

6 28 A.F. Vasil'ev, V.I. Murashka to each group a possibly empty set of subgroups of satisfying f( ) = ( f) for any isomorphism f :. Definition 4.2. Let θ be a subgroup functor and R be a subgroup of a group. We will call a subgroup H of the R -θ -subgroup if H θ ( H, R ). Let θ be the P-subnormal subgroup functor. Recall [18] that a subgroup H of a group is called P-subnormal in if H = or there is a chain of subgroup H = H0 < H1 <... < Hn = where Hi : Hi 1 is a prime for i = 1,..., n. O. Kramer's theorem [19, p.12] states Theorem 4.3. If every maximal subgroup of a soluble group is F-P-subnormal ( ) then is supersoluble. This theorem was generalized by Yangming Li, Xianhua Li in [9]. Theorem 4.4. A group is supersoluble if and only if every maximal subgroup of is F ( )-P -subnormal. It is well known that a group is nilpotent if and only if every maximal subgroup of is normal in. Theorem 4.5. A group is nilpotent if and only if every maximal subgroup of is F ( )-subnormal. Let θ be the conjugate-permutable subgroup functor. Recall [20] that a subgroup H of a x x group is called conjugate-permutable if HH = H H for all x. Corollary 4.4 [21]. A group is nilpotent if and only if every maximal subgroup of is F ( )-conjugate-permutable. Corollary 4.7 [8], [21]. If is a non-nilpotent group then there is an abnormal maximal subgroup M of such that =M F ( ). Theorem 4.8 The following statements for a group are equivalent: (1) is nilpotent; (2) Every abnormal subgroup of is F -subnormal subgroup of ; (3) All normalizers of Sylow subgroups of are F -subnormal subgroups of ; (4) All cyclic primary subgroups of are F -subnormal subgroups of ; (5) All Sylow subgroups of are F -subnormal subgroups of. Corollary 4.9. A group is nilpotent if and only if the normalizers of all Sylow subgroups of contain F ( ). Theorem 4.10 [22]. If a group is a product of two F()-subnormal nilpotent subgroups A and B then is nilpotent. Theorem Let a group =AB, where A and B are F()-subnormal supersoluble subgroups of. If [A,B] is nilpotent, then is supersoluble. The well known result states that a group is supersoluble if it contains two subnormal supersoluble subgroups with coprime indexes in. The following example shows this result will fail if we replace «normal» by «F()-subnormal». Let be the symmetric group on 3 letters. By theorem 10.3B [158, p. 173] there is a faithful irreducible F 7 -module V and the dimension of V is 2. Let R be the semidirect product of V and. Let A = V 3 and B = V 2 where p is a Sylow p-subgroup of, p {2, 3}. Since 7 1 (mod p) for p {2, 3}, it is easy to check that subgroups A and B are supersoluble. Since V is faithful irreducible module, F(R) = V. Therefore A and B are the F(R)-subnormal subgroups of. Note that R = AB but R is not supersoluble. Theorem Let A, B and C be a F()-subnormal supersoluble subgroups of a group. If indexes of A, B and C in are pairwise coprime then is supersoluble. Corollary Let A, B and C be a supersoluble subgroups of a group. If indexes of A, B and C in are pairwise coprime and F() A B C then is supersoluble. References

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Math-Net.Ru All Russian mathematical portal Wenbin Guo, Alexander N. Skiba, On the intersection of maximal supersoluble subgroups of a finite group, Tr. Inst. Mat., 2013, Volume 21, Number 1, 48 51 Use