JOURNAL OF ALGEBRA 195, 501]509Ž. 1997 ARTICLE NO. JA977034 Hall Subgroups and p-Separable Groups Zhaowei Du* Department of Mathematics, Uni¨ersity of Florida, Gaines¨ille, Florida 32611 Communicated by George Glauberman Received March 1, 1996 Let G be a finite group and p a set of primes. In this paper, some new criteria for p-separable groups and p-solvable groups in terms of Hall subgroups are proved. The main results are the following: Theorem. Gisap-separable group if and only if Ž.1 G satisfies Ep and Ep 9; Ž. p p 9 2 G satisfies Ep j Äq4 and Ep 9j Ä p4 for all p g , q g . Theorem. Gisap-separable group if and only if Ž.1 G satisfies Ep and Ep 9; Ž. 2 G satisfies Ep, q for all p g p and q g p 9. Q 1997 Academic Press INTRODUCTION The notation and terminology used in this paper are standard. The reader may refer towx 8 . Throughout the paper G denotes a finite group. p denotes a finite subset of the set of all primes. We let p Ž.G denote the set of all primes which divide the order of G. We say that G satisfies Ep if G has a Hall p-subgroup of G. We let Hallp Ž.G denote the set of all Hall p-subgroups of G. We call G a p-separable group if G has a composition ??? series 1 s K nnF K y110F F K F K s G, where each factor K iirK q1 Ž.0FiFny1 is either a p-group or a p 9-group. The relation between Hall subgroups and G has been studied quite extensively. This study can be traced back to the Schur]Zassenhaus Theorem which asserts that if G has a normal Hall p-subgroup, then G satisfies Ep 9. P. Hall made a systematic study for solvable groups and * E-mail address: [email protected]. 501 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 502 ZHAOWEI DU proved that G is solvable if and only if G satisfies Ep for all p. In fact, Hall proved a stronger result which states that G is solvable if and only if Ž. Gsatisfies Ep9 for all p g p G wx11 . Hall also conjectured: G is solvable Ž. if and only if G satisfies Ep, q for all p, q g p G wx11 . If G is a counterexample of minimal order to Hall's Conjecture, it is easy to see that G must be a non-abelian simple group. In fact, Hall himself verified Ž. that the alternating groups Ann G5 do not contradict the conjecture. Spitznagel, Jr. checked Hall's Conjecture for many, but not all, of the Chevalley groupswx 13 . Finally, Arad and Ward wx 4 proved Hall's Conjecture by using the classification of finite simple groups. They also proved that if Gsatisfies E29 and E39, then G must be solvablewx 4 . Guralnick studied finite simple groups which satisfy Ep9 for some prime pgpŽ.G and showed that such groups are very restrictedwx 9 . Arad and Fisman proved the same result inwx 3 . In w 3, 7 x , Arad and Fisman proved that finite simple groups satisfying Ep and Ep 9 are very restricted. They also generalized Hall's resultwx 11 and proved that a finite group G is k solvable if and only if G satisfies E , for some k such that 2 F k F <<G y 1, k where E means that G has a Hall Ä4p12, p ,..., pk-subgroup for all Ä4Ž. p12,p,..., pk:p G , where p12, p ,..., pkare distinct. Arad and Chillagwx 2 also proved the interesting result that a finite group having a Ä4 nilpotent Hall p-subgroup with 2 g p is p 00-solvable, where p s p R 2. Most of the above work concerns the solvability of finite groups. As we know, p-separable Ž.p-solvable groups are generalizations of solvable groups. A natural question to ask is whether we can use Hall subgroups to characterize p-separable Ž.p-solvable groups. For p-separable groups, the following result is well known: THEOREM wx8, Theorem 6.3.5 . Let G be a p-separable group. Then Ž.1 G satisfies Ep and Ep 9; Ž. p p 9 2 G satisfies Ep jÄq4 and Ep 9jÄ p4 for all p g and q g ; Ž. 3 G satisfies Ep, q for all p g p , q g p 9. What about the inverse of the above theorem? Namely, if G satisfiesŽ. 1 , Ž.2,and3is Ž. Gp-separable? In this paper we positively answer this question without using the classification of finite simple groups and give some new characterization for p-separable groups and p-solvable groups in terms of Hall subgroups. In fact, we prove a stronger result thatŽ. 1, 2 or Ž.1, 3 is sufficient for p-separability of a finite group G. In general, Ž. 2, 3 is not sufficient since PSLŽ. 2, 7 with p s Ä43 provides a counterexample. However, in the case where 2 f p and <p l p Ž.G <G 2, we prove that Ž.2, 3 is also sufficient for p-separability of a finite group G. In Section 1, we do not use the classification of finite simple groups. We prove thatŽ. 1, 2 implies that G is p-separable. We further obtain some HALL SUBGROUPS 503 related characterizations of p-separable, p-solvable, and p-solvable groups. In Sections 2 and 3, we do use the classification of finite simple groups. We prove thatŽ. 1, 3 implies that G is p-separable. We further obtain some related characterizations of p-separable, p-solvable, and p-solvable groups. In Section 3, we prove thatŽ. 2, 3 does imply that G is p-solvable when 2fpand <p l p Ž.G <G 2. In the following, we use CFSG to refer to the classification of finite simple groups. 1. NEW CHARACTERIZATION WITHOUT CFSG In this section, without using CFSG, we give some new characterizations for p-separable groups, p-solvable groups, and p-solvable groups. THEOREM 1. Gisap-separable group if and only if Ž.1 G satisfies Ep and Ep 9; Ž. p p 9 2 G satisfies Ep jÄq4 and Ep 9jÄ p4 for all p g and q g . Proof. Ž.« follows immediately from Hall's Theorem, e.g.w 8, Theorem 6.3.5x . Let us prove the other direction. Ž.¥Suppose Theorem 1 is false and let G be a minimal counterexam- ple. Without loss of generality we may assume p Ž.G s p j p 9. Namely both p and p 9 are subsets of p Ž.G and p 9 is the complement of p in pŽ.G. We divide the proof into several steps. Ž.1 Gis a non-abelian simple group. Otherwise, there is a non-trivial proper normal subgroup K of G. Since the assumption is inherited by normal subgroups and quotient groups of G, K and GrK are both p-separable, and therefore G is p-separable. This is a contradiction. ThusŽ. 1 holds. Ž. Ž. Let A g Hallp G , B g Hallp9 G . It is easy to see that G s AB.By ab Burnside's pq Theorem, either <<p G 2or <p9 <G2. Without loss of generality, we assume <<p 9 G 2. Ž.2 Bis not a solvable group. Assume otherwise. Let K be a minimal normal subgroup of B. Then a <<Ksqfor some prime q and some positive integer a. By assumption, Ž.<<p 9 there exists M g Hallp jÄq4 G . G 2 implies M z G. By Sylow's Theo- rem, K is contained in some conjugate of M, so that we may assume KFM. By Hall's Theorem, e.g.wx 8, Lemma 6.4.2 , G s BM. Hence G BM M M 1/² K:²sK:²sK:²:M:zG 504 ZHAOWEI DU implies that M contains a non-trivial proper normal subgroup ² K G : of G. This is contrary toŽ. 1 , and thus Ž. 2 holds. Ž.32gp9. Otherwise, 2 g p. Then B is of odd order and solvablewx 6 . This contradictsŽ. 2 . Thus Ž. 3 holds. Ž.4<<ps1. We let p s Ä4p . Otherwise, <<p G 2. By the same arguments ofŽ. 2 and Ž. 3 , we will get 2gp. This contradictsŽ. 3 , hence Ž. 4 holds. Let p 9 Ä42 q , q ,...,q , r 2, M Hall Ž.G ,1 i r. s s 12 riG g pjÄqi4F F Ž.Ž.5OM OM Ž. 1 for all 1 i r. p9 iqis i s F F Otherwise, assume OMŽ./1 for some i. By Sylow's Theorem OMŽ. qii p9i OMŽ.is contained in some conjugate of B, so that we may assume sqii Ž. OMp9 i FB. By Hall's Theorem, e.g.wx 8, Lemma 6.4.2 , G s BMi, and therefore GMiBB 1/¦;¦;¦;OMp9Ž.ipsOM9 Ž.ipsOM9 Ž.i:BzG implies that B contains a non-trivial proper normal subgroup ²K G:of G. This contradictsŽ. 1 , so that Ž. 5 holds. Ž.6 Final contradiction. Ž. Since Mjj2 F j F r is of odd order, M is p-constrained. On the other hand, bywx 1, Theorem 8.1.2 , G is also p-stable. By Sylow's Theorem, Ž. without loss of generality, we may assume P s A F Mj 1 F j F r .Ac- cording towx 8, Theorem 8.1.11 , we have M OMNZJPŽ.Ž.