On subgroups of ZJ type of an F-injector for fitting classes between Ep*p and Ep*Gp

Publicacions Matemàtiques, Vol 38 (1994), 411-425 . ON SUBGROUPS OF ZJ TYP E OF AN . -INJECTOR FORFITTING CLASSE S BETWEEN ~ p*p AN D A . MARTÍNEZ PASTOR (* ) A bstract Let G be a finite group and p a prime . We consider an 1injecta r K of G, being 1 a Fitting class between ~p* p and ~ P* ep, and w e study the structure and normality in G of the subgroups ZJ(K ) and ZJ* (K), provided that G verify certain conditions, extendin g some results of G . Glauberman (A characteristic subgroup of a pstable group, Canad . J . Math . 20 (1968), 555-564) . 1 . Introduction and notatio n In this paper we consider a finite group Gverifying certain condition s of stability and constraint, and we study the structure and normalit y in G of the subgroups ZJ(K) and ZJ*(K), being K and 1-injector o f G and 1 a Fitting class such that L~ p*p Ç C extending som e results of Glauberman [61 . A11 groups in this paper are assumed to be finite . Given a fixed prim e p, e p will denote the class of all p -groups, ~ p* , the class of all p* -groups , L~p * p the class of all p*p -groups and that of all p* - by - p -groups . The corresponding radicals in a group G are denoted by Dp(G) , O p . (G) , G p * p (G) and O p . , p (G) respectively . Far all definitions we refer toBende r [ 3 1 The notation for Fitting classes is taken from [1• The remainder o f the notation is standard and it is taken mainly from [7] and [8] . I n particular, E(G) is the semisimple radical of G and F* (G) = F(G)E(G ) the quasinilpotent radical of G . If H is a subgroup of G, C'¿(H) is th e generalized centralizer of H in G (see [31)• Note that C Z( F* (G} ) < ( *)Work supported by the CICYT of the Spanish Ministry of Education and Science , project :PB90-o414-0O3-ol . 412  A . MARTÍNEZ PASTO R F (G), in every group G . A group G is said to be 91-constrained i f C G (F(G)) Ç F(G), that is, if E(G) = 1 . Moreover, 7r(G) is the set of primes dividing the order of G, d(G ) is the maximum of the orders of the abelian subgroups of G, 2t(G) i s the set of all abelian subgroups of order d(G) in G and J(G) is th e subgroup generated by 91(G), that is, the Thompson subgroup of G . W e set ZJ(G) = Z(J(G)) . In [fi] G . Glauberman proves his well-known ZJ-Theorem and als o introduces the subgroup ZJ*(P) proving the following : "Let p be a n odd prime and let P be a Sylow psubgroup of a group G . Suppos e that C G (O p (G)} ç ap (G) and that SA(2,p) is not involved in G . The n ZJ*(P) is a characteristic subgroup of G and C G (ZJ* (P)} Ç ZJ * (P)" . On the other hand, Arad and Glauberman study in [2] the structur e and normality of the subgroup ZJ(H), H being a Hall 7r-subgroup of a 7 rsoluble group G with abelian Sylow 2-subgroups and 0,,(G) = 1 . Some related results were obtained by Arad in [1], by Ezquerro in [5 ] and by Pérez Ramos in [11] and [12] . Here we study the structure of the subgroups ZJ(K) and ZJ*(K ) where K is an 1-injector of G, being 1 a Fitting class such that 4 . p Ç Ç and we obtain that it depends only of G . Also, we obtai n some analogous to Glauberman's ZJ and ZJ* Theorems for such Fittin g classes . Recall that such a Fitting class is dominant in the class o f all finite groups, so every finite group G has a unique conjugacy clas s of 1-inj ectors (see [10]) . Moreover, for such every finite group i s constrained in the sense of [9] (see [3]) . In the following will be a Fitting class such that ~p p C C e p * ep . 2 . Preliminary result s Remark 1 . Let K be an 1 -injector of a group G . By [10] we know tha t K = (O p . (G)P) 1 where P is a Sylow p -subgroup of G . Moreover, 4p* (K) = p p* (G), s o 4 p ► (K) = Cp► (G) and Qp► (F(K)} = ap► (F(G)} . On the other hand , since F*(G) Ç K, we have E(K) = E(G) . Remark 2 . Suppose that K is an ~p* C~p-group, that is, K = ap* (K ) S where S i s a Sylow psubgroup of K . Since [Op .(K),Op(K)] = 1, it is clear that K ON SUBGROUPS OF ZJ TYPE  41 3 acts nilpotently on Qp (K), i .e . K = CK (Cp (K)} . In particular, we ca n deduce that CK(E(K)Op,(F(K))) = CK(F*(K)) < F(K) . Lemma 2 .1 . Let G be a group and let K be an e P •C7 P -subgroup of G containin g F*(G) . Then 7r(ZJ(K)) Ç 'n-(F(G)) = 7r(F(K)) . Moreover if the prim e p belongs to 1r(F(G)) then p E 7r(ZJ(K)) . Proof . • Since 7r(F(K)) = 7r(Z(F(K)) and Z(F(K)) < C G (F*(G)) < F(G) , the first statement can be easily obtained . On the other hand if p E 7r(F(G)) and P is a Sylow p-subgroup of K we have 1 Z(P)nO P (K ) Z(K) < ZJ(K) since K =PO P • (K), and so the result holds . ■ Lemma 2 .2 . Let G be a group and let K be an P •e P -subgroup of G containin g Op(G) . Let B be a nilpotent normal subgroup of G and let A be an y nilpotent subgroup of K . Then AO P (B) is nilpotent . Proof : By the Remark 2 A acts nilpotently on C p (B) Ç p p (K), so the resul t follows . ■ Next we will deal with the subgroup ZJ*(K) of an arbitrary group K and its properties : Defi .nition 2 .3 . [5] . For any group K define two sequences of characteristic subgroups o f K as follows . Set ZJ° (K) = 1 and Ko = K . Given ZJ z (K) and K i , i ~ O, let ZJ i+1 (K) and Kz + 1 the subgroups of K that contain ZJ a (K ) and satisfy : ZJ z+l (K)/ZJ i (K) = ZJ ( K Z/ ZJZ ( K ) ) K i+1I zJZ ( K ) = c xilZJ i (K)( ZJz+l ( K )l zJ2 ( K )) • Let n be the smallest integer such that Z J n (K ) = Z Jn+ x (K ) , the n ZJ n (K) = ZJ n+r (K) and K n = K n + r for everyn ~ O . Set ZJ*(K) = ZJ n (K) and K * = Kn . 414  A . MARTÍNEZ PASTO R Example . In general, the subgroups ZJ(K) and ZJ*(K) of a group K are different . To see this, we can consider, as an example, the group K _ [Q8 x C3] S3 generated by the elements a, b, c, x, y with the following relations : a 4  a z =b z a ó a -1 C 3= 1 a `= a b c =b, x32 J 21 , xy= y -1 , a s = ba, b s = a -1 , c s = c, a y = b, b y = a, c y = c -1 . Then we can get check that d(K) = 18, Z(K) = Z(Q 8 ) = (a 2 ), ZJ(K) _ Z(Q 8 ) x C 3 , K 1 = [Q $ x C 3 ] (x) = J(K) and ZJ*(K) = ZJ 2 (K) = K 2 = [Q 8 xC 3 ] . Remark 3 . For every group K : i) ZJ(K i /ZJ i (K)) = ZJ(K i + l /ZJ Z (K)) = Z(K i + l /ZJ i (K)), fo r every i > O . ii) Z(K Z ) Z(K2+1), for every i > O . Lemma 2 .4 . For any group K and for every i > O : i) ZJ i (K) is nilpotent . ii) F(K i /ZJ z (K)) = F(K i )/ZJ i (K) . Proof : i) By induction on i, assume that ZJ i (G) is nilpotent, for ever y group G . By ([5, Prop . II 3 .6]) we have that ZJ i+l (K)/ZJ 1 (K) = ZJ i (K l /ZJ l (K)), so this is a nilpotent group . Now, by the previou s remark, ZJ l (K) = ZJ(K) < Z(K 1 ) < Z(K i ), and ZJ i+l (K) < K ti , hence ZJ z+l (K) is nilpotent . ii) By induction on i . The assertion is clear for i = O . Assume no w that F(K Z /ZJ i (K)) = F(K Z )/ZJ i (K) . We have : F ( K i+ 1 / z Ji+ 1 ( K )) = F ( K i+ 1 I ZJi ( K )l zJ2+ l ( K )I ZJZ ( K ) ) and since ZJ i+l (K)/ZJ i (K) = Z(K Z + 1 /ZJ i (K)), it follow s F ( K Z+iI zJZ ( K )I zJZ+i ( K )I ZJZ ( K )) _ F ( K z+1I zJZ ( K ))l zJ2+l ( K )l zJi ( K ) But applying the inductive hypothesis we have : F ( K i+1I zJZ ( K )) =F ( K zI zJZ ( K )) n K 2+1/ ZJZ ( K ) = F ( K z)l ZJ2 ( K ) n K Z+1l zJ2 ( K ) = F ( K i+il zJz ( K ) and so we can conclude that F(K i + l /ZJ i+l (K)) =F(Ki+l)/ZJi+l(K) ■ ON SUBGROUPS OF ZJ TYPE  41 5 3 . The structure of the ZJ-subgrou p and the ZJ * -subgrou p In this section we will study the structure of the subgroups ZJ(K) an d ZJ *(K) being K an ~p* ep-subgroup of a group G containing Qp (G) an d satisfy :ing that C p* (K) = G p* (G), properties that hold for an 1-inj ecto r of G, as we have seen . Theorem 3 .1 . Let G be an 91-constrained group and let K be an p ,»C7 P -subgroup o f G containing O P (G) and such that O p . (K) = O p . (G) . Assume that a t least one of the following conditions hold : i) 0 p ,(F(G)) ZJ(K) , ii) F(G) is abelian , iii) d(K) is odd and 0 2 (G) is abelian . Then : a) {O P (A)CA E 9t(K)} = 2I(O P (K)) . b) O p (ZJ(K)) = ZJ(O p (K)) . c) {O ' (A)IA E 9t(K)} _ 21(9p•(G)) . d) O p i (ZJ(K)) = ZJ(O P • (G)) . In particular, if we assume O P ,(F(G)) < ZJ(K) then for every A E 2((K) O P A(A) = O p i(ZJ(K)) = O P A(F(G)) . Moreover the prime numbers divisors of d(K), ZJ(K)1, IF(K)1 an d 1F(G)i coincide . Proo f : Let A E 1(K) . Since F*(G) Ç K we know that E(K) = E(G) = 1 , so K is an 91-constrained group . Leading from our assumptions we can obtain that AF(G) is nilpotent (if we assume i) Lemma 2 .2 applies ; if we assume ii} or iii) Proposition 1 of [2] applies) . Moreover, sinc e G p* (K) = G p* (G) we have Cp' (F(K)} = Cp' (F(G)} . a) Let A E 1(K) . Since AF(G) is nilpotent O p (A) centralize s C p' (F (G)} and so applying Remark 2 we obtain O p (A) ç C K (D p ► (F(K)}} Ç F(K ) so O p (A) D p (K) . Let B E 21(0 p (K)} . Since ADp (K) is nilpotent by Lemma 2 .2, C p' (A ) centralizes G p (K), so C p' (A)B is an abelian subgroup of K and the n 10 p ,(A)Bi ~ I A I = 10 p ,(A)O p (A)I . 416  A . MARTÍNEZ PASTO R Hence d(Op(K))  1 O P (A) I . Since O p (A)  O P (K) the equalit y d(O p (K)) = 1 O P (A) 1 holds . Thus, for every B E 2((Op(K)), O P ,(A) x B E I[(K) . So we hav e {O P (A)CA E 91(K)} = 1(O p (K)) . b) This follows easily from a) : O P (ZJ(K)) = O P (n{ASA E 91(K)} ) = n{O P (A)CA E%(K)} = ZJ(O P (K)) . c) Let A E 91(K) . By a) we know that O p (A) < O P (K) . On the othe r hand, since K is an P • C7p-group we have O P A(A) < O P (K) = O P • (K) = O P .(G) . Let B E 9 , l(Op• (G)) . Since [O(G), O P (K)] = 1, Op(A) centralizes B so Op(A)B is an abelian subgroup of K and the n lO p ,(A)B1  IA1 = 10 p ( A)O p ,(A)I . Hence d(O P • (G) ) < I 0 p , (A) I . Since O p i (A) c 0 p . (G) it follow s d(O p . (G)) = I O P , (A) 1 . Therefore, for every B E 2 (O p • (G), O p (A) x B E 2((K) . This proves c) . d) This follows from c) as in b) . If we assume O P A(F(G)) < ZJ(K) then it is clear that Opi(ZJ(K)) = O P ,(F(K)) = O p ,(F(G)) . Let A E %(K) . Since ZJ(K) = fl{AIA E 9((K)} and AF(G) is nilpotent we obtain that O P ,(A) C G (F(G)) G F(G) and so the equality O P A(F(G)) = O p i(ZJ(K)) = O P A(A) holds . Now since F* (G) < K we can apply Lemma 2 .1 and our assumption s to obtain 7r(ZJ(K)) = 7r(F(G)) = 7r(F(K)) . Moreover, if A E %(K ) it is clear that 7 r(ZJ(K)) C 7r(A) = 7r(d(K)) . On the other hand, if q is a prime number such that q p and q E Ir(A), then q E 7r(F(G)) , by the foregoing assertion . Finally, if we assume that p E 7r(A), the n p E 7r(F(K)) = 7r(F(G)) because of a), and so the result follows . ■ Corollary 3 .2 . Let G be an 91-constrained group, H an ~p* ep-injector of G and K = its associated 1-injector of G . lf one of the following condition s holds : i) G p ,(F(G)} Ç ZJ(K) , ii} F(G) is abelian , iii) d(K) is odd and 0 2 (G) is abelian, ON SUBGROUPS OF ZJ TYPE  41 7 then ZJ(K) = ZJ(O p .(G)) x ZJ(O p (H)) = ZJ(H) . So, in particular, ZJ(K) Boes not depend on the Fitting clas s Proof : Given A in 21(H), by Remark 2 we see that O p (A) O P (H) = O P (K) . On the other hand, due to the structure of the injectors considered here , one has O PA(A) < O P (H) = O p . (H) = Op .(G) < K . Therefore 2t (H) = 9I(K) . Then apply Theorem 3 .1 parts b) and d) to the subgroups H an d K . ■ Corollary 3 .3 . If G is an 9t-constrained group and K and 1-injector of G such tha t O p i (F(G)) Z(K), then K = O P ,(F(G)) x P where P is a Sylow p-subgroup of G . In particular , 2((K) _ {O p ,(F(G))AIA E Il(P)} . Proof : Since K= PG P* (G), P a Sylow p -subgroup of K and Gp► (F(G)} < Z(K), due to 6 .11 in [3], we can write [P, G p* (G)] = 1 . Now by g tconstraint, K is nilpotent and hence it is an 4► 6p-injector of G (se e [10] } ; therefore P is a Sylow p -subgroup of G and K = Q p ► (F(G)} x P . ■ Our next goal is to study the structure of the ZJ * -subgroup . Theorem 3 .4 . Let G be an 91-corástrained group . Let K be an p* ep-subgroup o f G containing G p (G) and such that Q p* (K) = Qp* (G) . Assume tha t 0p► (F(G)} Ç ZJ(K) . Denote P = Gp (K) . Then for every i ~ 1 , G P ► (ZJ i (K)} = Gp ► (F(Kz )} = Gp► (F(G)), K i is a nilpotent group an d O P( ZJ - Z ( K )) = ZJ Z (P) O p( K i) = P i with the notation given in Definition 2 .3 . In particular Op(ZJ*(K)) _ ZJ*(P), O P (K * ) = P * an d ZJ*(K) = ZJ*(P) x O p ,(F(G)) . 418  A . MARTÍNEZ PASTO R Proof : Since O P ,(ZJ(K)) < O P ,(ZJ i (K)) < O P ,(F(K Z )) c O P ,(F(K)) = Opi (F(G)), the first statement is clear . Notice that Op,(F(G))<ZJ(K)<Z(Kl), so O p . (K 1 ) < CG(F(G)) G F(G) . Hence  (K 1 ) = O P ,(F(K l )) < Z(K 1 ) and K l is a nilpoten t gorup . Now apply that for every i > 1, K i < K l . We will prove that O p (ZJ Z (K)) = ZJ i (P) and O p (K Z ) = P i by induction on i . By Propositiom 3 .2 we have ZJ(P) = O p (ZJ(K)) . O n the other hand P = O p (K) centralizes Op,(ZJ(K)), so Cp(ZJ(P) ) C K (ZJ(K)) and then we obtai n O p (K l ) = P n K l = P n C K (ZJ(K)) = Cp(ZJ(P)) _ Thus, the statement is clear for i , = 1 . Now suppose that ap(ZJ i (K)} = ZJ i (P) and Gp(Ki) = Pi . Applyin g Lemma 2 .4 and the fact that ap ► (F(K i )} = Op►(ZJi(K)} , we get tha t K i 'ZJ i (K) = F(K i )/ZJ i ( K) is a p-group . Then it follows tha t K i / ZJ 2 (K) = P Z ZJZ ( K )I ZJZ ( K ) = P Zl zJ2 ( K ) n P i = P Zl zJ2 ( P ) by the inductive hypothesis . Thu s ZJ 2+1 (K)/ZJ Z (K) = ZJ(K i /ZJ Z (K)) ZJ(P i /ZJ i (P) ) =ZJi+l ( p )/ ZJi ( p ) and since ZJ i +1 (K) = ZJ i (K)(ZJ i +l (K) n P i ) we can conclud e op (ZJ z +1 (K)) = ZJ' +1 (K) n O(K) = ZJi +1 (K) n P i = zJ' 1 (P) . Now we will prove that Gp (K i + 1 ) = P i +1 . It is clear that O(K 1 )Ç Op(Ki) = P i an d [O(K 1 ), ZJ '+1 (P)] Ç [O(K+ 1 ), Z,P +1 (K) ] ap(Ki+1) n ZJ i (K) = ZJ i (P) . Hence by the definition of P i + 1 it follows that O p, ( K i +1 ) Ç Pi+1 . On th e other hand, P i + l P i Ç Ki and since D p ► (F(G)} Ç ZJ(K) Ç Z(K i ) , we hav e [Pi,ZJ'(K)] = [ZJ i (P) ZJ z (K) . Thus, by the definition of K i + 1 we obtain Pi+1 Ç Ki +1 . Now, sinc e Q p ( K i +1 ) is the Sylow p-subgroupof I? i+l the result follows . ■ ON SUBGROUPS OF ZJ TYPE  41 9 Corollary 3 .5 . Let G be an 92-constrained group . Let H be an ~p• C5p-injector of G and assume that Op,(F(G)) < ZJ(H) . Let K = H l be an 1-injector o f G . Then ZJ*(K) = O P ,(F(G)) x ZJ*(O p (H)) = ZJ*(H) . In particular, ZJ*(K) loes not depend o n Proof : Because of Corollary 3 .2 we have ZJ(K) = ZJ(H) . Now Theorem 3 . 4 is applied, keeping in mind that Op(K) = O P (H) . ■ 4 . The normaiity of the ZJ-subgrou p and the ZJ *- subgrou p In this section we prove some results related to the normality of th e ZJ-subgroup and the normality and self-centrality of the Z J *- subgrou p of an 1-inj ector K of a group G, provided that G verifies certain conditions of stability . Concretely, we will use the following version of pstability : Definition 4 .1 . A group G is said to be p -stable if whenever A is a subnormal psubgroup of G and B is a p -subgroup of N G (A) satisfying [A, B, B] = 1 , then B Ç CptN G ( A ) mod C G (A)) . Proposition 4 .2 . Let G be a p -stable group . Let K be an L~p * C~p - subgroup of G containin g the ~p* p -radicad of G, Dp* p (G) . If N is an abelian normal subgroup o f K then N oo G and N Ç F(G) . In particular ZJ(K) Ç F(G) . . Proof : First notice that C)p*p(G) Ç K implies O p* (K) = D p* (G) (see [3 , 4 .22]} . Thus, D p ,(N) < p p* (G) < Gp*p(G)  K, and so Gp►(N ) () p *p (G) . On the other hand, it holds [O(G), G p (N), C) p (N)] = 1 and so applying the p -stability of G we have : O P( N ) C G( O p( G ))l c G( O P( G )) O P( G I C G( O p( G )) ) = C c( O P( G ))l c c( O p( G ))