- Email: [email protected]
Hall Subgroups and π-Separable Groups
195, 501]509 Ž1997. JA977034
JOURNAL OF ALGEBRA ARTICLE NO.
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. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9 ;
Ž2.
G satisfies Ep j q4 and Ep 9j p4 for all p g p , q g p 9.
Theorem. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9 ;
Ž2.
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 to w8x. 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 Hall p 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 n F K ny1 F ??? F K 1 F K 0 s G, where each factor K irK iq1 Ž0 F i F n y 1. 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 G satisfies Ep9 for all p g p Ž G . w11x. Hall also conjectured: G is solvable if and only if G satisfies Ep, q for all p, q g p Ž G . w11x. 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 that the alternating groups A n Ž n G 5. do not contradict the conjecture. Spitznagel, Jr. checked Hall’s Conjecture for many, but not all, of the Chevalley groups w13x. Finally, Arad and Ward w4x proved Hall’s Conjecture by using the classification of finite simple groups. They also proved that if G satisfies E2 9 and E39 , then G must be solvable w4x. Guralnick studied finite simple groups which satisfy Ep9 for some prime p g p Ž G . and showed that such groups are very restricted w9x. Arad and Fisman proved the same result in w3x. In w3, 7x, Arad and Fisman proved that finite simple groups satisfying Ep and Ep 9 are very restricted. They also generalized Hall’s result w11x and proved that a finite group G is solvable if and only if G satisfies E k , for some k such that 2 F k F < G < y 1, where E k means that G has a Hall p1 , p 2 , . . . , pk 4 -subgroup for all p1 , p 2 , . . . , pk 4 : p Ž G ., where p1 , p 2 , . . . , pk are distinct. Arad and Chillag w2x also proved the interesting result that a finite group having a nilpotent Hall p-subgroup with 2 g p is p 0-solvable, where p 0 s p R 24 . Most of the above work concerns the solvability of finite groups. As we know, p-separable Žp-solvable. groups are generalizations of solvable groups. A natural question to ask is whether we can use Hall subgroups to characterize p-separable Žp-solvable. groups. For p-separable groups, the following result is well known: THEOREM w8, Theorem 6.3.5x. Ž1. Ž2. Ž3.
Let G be a p-separable group. Then
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9; G satisfies Ep, q for all p g p , q g p 9.
What about the inverse of the above theorem? Namely, if G satisfies Ž1., Ž2., and Ž3. is G p-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 that Ž1, 2. or Ž1, 3. is sufficient for p-separability of a finite group G. In general, Ž2, 3. is not sufficient since PSLŽ2, 7. with p s 34 provides a counterexample. However, in the case where 2 f p and
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 that Ž1, 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 that Ž2, 3. does imply that G is p-solvable when 2 f p and
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. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9;
Ž2.
G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9.
Proof. Ž«. follows immediately from Hall’s Theorem, e.g. w8, Theorem 6.3.5x. Let us prove the other direction. Ž¥. Suppose Theorem 1 is false and let G be a minimal counterexample. 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 p Ž G .. We divide the proof into several steps. Ž1. G is 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. Thus Ž1. holds. Let A g Hall p Ž G ., B g Hall p 9Ž G .. It is easy to see that G s AB. By Burnside’s p aq b Theorem, either
504
ZHAOWEI DU
implies that M contains a non-trivial proper normal subgroup ² K G : of G. This is contrary to Ž1., and thus Ž2. holds. Ž3. 2 g p 9. Otherwise, 2 g p . Then B is of odd order and solvable w6x. This contradicts Ž2.. Thus Ž3. holds. Ž4.
¦
G
;s¦O
p9
Ž Mi .
Mi B
;s¦O
p9
Ž Mi .
B
;: B z G
implies that B contains a non-trivial proper normal subgroup ² K G : of G. This contradicts Ž1., so that Ž5. holds. Ž6. Final contradiction. Since M j Ž2 F j F r . is of odd order, M j is p-constrained. On the other hand, by w1, Theorem 8.1.2x, G is also p-stable. By Sylow’s Theorem, without loss of generality, we may assume P s A F M j Ž1 F j F r .. According to w8, Theorem 8.1.11x, we have M j s Op9 Ž M j . NM jŽ Z Ž J Ž P . . . ,
for all 2 F j F r .
By Ž5. Op9Ž M j . s 1, hence ZŽ J Ž P ..e } M j for all 2 F j F r. Let X s NG Ž ZŽ J Ž P .... Then < G : X < s 2 a, for some a. Let M1 s PQ for some Q g Syl 2 Ž G .. Then G s XQ. Therefore 1 /² Z Ž J Ž P . . G: s² Z Ž J Ž P . . X Q: s² Z Ž J Ž P . . Q: : M1 z G implies that M1 contains a non-trivial proper normal subgroup ² ZŽ J Ž P .. G : of G. This contradicts Ž1., so that Ž6. holds. The proof of Theorem 1 is complete. THEOREM 2. G is a p-sol¨ able group if and only if Ž1. Ž2.
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j s for all q g p 9 and s : p .
Proof. Ž«. Let G be a p-solvable group, and we prove that G satisfies Ž1. and Ž2.. By Theorem 1, G satisfies Ž1.. G also satisfies Ep j q4.
HALL SUBGROUPS
505
Now all we need to do is to show that G satisfies Ep 9j s . For every s : p , we show that G is p 9 j s-separable. Let K be any chief factor of G. Since G is p-solvable, K is either a p 9-group or a p-group for some p g p . If p g s , then K is p 9 j s-group. If p g p R s , then K is a p 9 j s 4 9-group. Therefore G is p 9 j s-separable and G satisfies Ep 9j s . By Hall’s Theorem, e.g. w8, Theorem 6.3.5x, G satisfies Ep 9j s . Ž¥. By Theorem 1, G is p-separable. Let K be any chief factor of G. Then K is either a p 9-group or a p-group. If K is a p-group, by Ž2., K satisfies Es for all s : p Ž K .. By Hall’s Theorem, e.g. w8, Theorem 6.4.5x, K is a solvable group. This forces K to be a p-group for some p g p . This means that G is p-solvable. Hence the proof of Theorem 2 is complete. COROLLARY 1. G is a p-sol¨ able group if and only if Ž1. Ž2. Ž3.
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9; G has a Hall p-subgroup which is sol¨ able.
Proof. Corollary 1 follows easily from Theorem 1. When p s p4 , we have the following simple conditions. COROLLARY 2. G is a p-sol¨ able group if and only if Ž1. Ž2.
G satisfies Ep9; G satisfies Ep, q for all q g p Ž G ..
Proof. Corollary 2 is a special case of Theorem 2. Remarks. Ž1. By Feit]Thompson’s Odd Order Paper w6x, when 2 f p , Assumption Ž3. in Corollary 1 can be omitted. Ž2. By a result w5x which says that G is solvable if and only if G satisfies E2, p for all p, when p s 2, Assumption Ž1. in Corollary 2 can be omitted.
2. NEW CHARACTERIZATIONS WITH CFSG In this section, we provide some new characterizations of p-separable and p-solvable groups by using CFSG. THEOREM 3. G is a p-separable group if and only if Ž1. Ž2.
G satisfies Ep and Ep 9; G satisfies Ep, q for all p g p and q g p 9.
Proof. Ž«. follows immediately from Hall’s Theorem, e.g. w8, Theorem 6.3.5x. Let us prove the other direction.
506
ZHAOWEI DU
Ž¥. Suppose Theorem 3 is false and let G be a minimal counterexample. To make our proof symmetric in p and p 9, we assume for convenience that p Ž G . s p j p 9, i.e., p 9 is the complement of p in p Ž G .. We divide the proof into several steps. Ž1. G is 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 p-separable. Therefore G is p-separable, and this is a contradiction. Thus Ž1. holds. Without loss of generality, we assume 2 g p . We let A g Hall p Ž G ., B g Hall p 9Ž G .. It is easy to see that G s AB. Ž2.
A r with r G 5 a prime and A , A ry1.
ŽII. M11 and either A is solvable or A , M10 . ŽIII. M23 and either B is Frobenius of order 11 ? 23 or B is cyclic of order 23 and A , M22 . ŽIV. PSLŽ2, q . where either q g 11, 29, 594 and A , A 5 or 3 - q k 1Ž4. and A is solvable. ŽV. PSLŽ r, q . with r an odd prime such that Ž r, q y 1. s 1 and either G , PSLŽ5, 2. and < B < s 5 ? 31 or A is a maximal parabolic such that PSLŽ r y 1, q . is involved in A. In particular B is either cyclic or Frobenius. This follows from w3, Theorem 1.1x which classifies all finite non-abelian simple groups which can be written as a product of two proper Hall subgroups with relative prime orders. Ž4. A is not solvable. Assume A is solvable. In this case both the Hall p-subgroup and the Hall p 9-subgroup are solvable. Hence G satisfies Er, s for all r, s g p Ž G .. By Hall’s Conjecture, proved by Arad and Ward w4x, G is solvable. This contradicts Ž1. and shows Ž4.. Ž5. G is not in Case ŽI. and G is not in Case ŽII.. This follows directly from Ž2. and Ž4.. Ž6. G is not in Case ŽIII.. Otherwise, by Ž2., we have G , M23 and B is Frobenius of order 11 ? 23. Then, by assumption, M23 satisfies E3, 23 . But, since the Hall 3, 234 subgroups of G are nilpotent, this contradicts the fact that 234 is a simple prime graph component w14x. Thus Ž6. holds. Ž7. G is not in Case ŽIV..
HALL SUBGROUPS
507
Otherwise, by Ž4., G , PSLŽ2, q . where q g 11, 29, 594 and A , A 5 . If q s 11, then we have p 9 s 114 . This contradicts Ž2.. If q s 29, then we have p 9 s 7, 294 . This implies that PSLŽ2, 29. satisfies E3, 29 . But, since the Hall 3, 294 -subgroups of G are nilpotent, this contradicts the fact that 294 is a simple prime graph component w14x. If q s 59, then we have p 9 s 29, 594 . This implies that PSLŽ2, 59. satisfies E5, 59 . But, since the Hall 5, 594 -subgroups of G are nilpotent, this contradicts the fact that 594 is a simple prime graph component w14x. Ž8. G is not in Case ŽV.. Otherwise, assume first G , PSLŽ5, 2. and < B < s 5 ? 31. Then PSLŽ5, 2. satisfies E3, 31 , and therefore GLŽ5, 2. satisfies E3, 31. But this contradicts w13, Corollary 2.1.6x. So we can assume G , PSLŽ r, q . with r an odd prime such that Ž r, q y 1. s 1 and A is a maximal parabolic such that PSLŽ r y 1, q . is involved in A. If q ) 3, let X be a Borel subgroup of G such that X F A, and X s PH, where P is a Sylow p-subgroup of G and H is a Cartan subgroup of G in X. By assumption G satisfies Ep, r for all r g p 9, so there is some Sylow r-subgroup R of G such that PR is a subgroup of G. By w13, Theorem 5.3x, R F X. This implies r g p for all r g p 9 which contradicts Ž2.. So we may assume q F 3. In this case q s 2 or 3. Assume first q s 2, G , PSLŽ r, 2., r G 3 a prime. In this case PSLŽ r, 2. s SLŽ r, 2. s GLŽ r, 2.. Let p g p 9 by assumption, G satisfies E2, p . Reference w13, Theorem 2.3.2x says that p s 3 Žalso r F 5.. This implies
G satisfies Ep and Ep 9;
Ž2.
G satisfies Es j q4 for all s : p and q g p 9 l p Ž G ..
Proof. We only need to prove the ‘‘if’’ part. By Theorem 3, G is p-separable. Let K be any chief factor of G. Then K is either a p-group or a p 9-group. To finish the proof, we only need to consider the case where K is a p-group and prove that K is solvable. For each s : p , there
508
ZHAOWEI DU
exists some q g p 9 l p Ž G .. By Ž2., G satisfies Es j q4. Hence K satisfies Es . So K has Hall s-subgroups for all s : p . It then follows from Hall’s Theorem, e.g. w8, Theorem 6.4.5x, that K is solvable. Therefore the proof is complete. COROLLARY 3. G is a p-sol¨ able group if and only if Ž1. Ž2. Ž3.
G satisfies Ep and Ep 9; G satisfies Ep, q for all p g p and q g p 9; G has a Hall p-subgroup which is sol¨ able.
Proof. We only need to prove the ‘‘if’’ part. By Theorem 3, G is p-solvable. Let K be any chief factor of G. Then K is either a p-group or a p 9-group. To finish the proof, we only need to consider the case that K is a p-group and prove that K is solvable. Let H be a solvable Hall p-subgroup of G. Since K is isomorphic to some section of H, K is solvable as we desired. The proof is complete. Remark. By Feit]Thompson’s Odd Order Paper w6x, when 2 f p , Assumption Ž3. in Corollary 3 can be omitted.
3. FURTHER DISCUSSION From Sections 1 and 2, we know that either Ž1, 2. or Ž1, 3. of Hall’s Theorem, e.g. w8, Theorem 6.3.5x, implies the p-solvability of groups. Therefore we might expect that Ž2, 3. also implies the p-solvability of groups. Unfortunately, it is not so. PSLŽ2, 7. provides a counterexample with p s 34 which satisfies E3, 2 and E3, 7 , but PSLŽ2, 7. is a finite non-abelian simple group. However, we can prove the following theorem: THEOREM 5. Ž1. Ž2.
Assume 2 f p and
Ep j q4 and Ep 9j p4 for all p g p l p Ž G . and q g p 9 l p Ž G .; Ep, q for all p g p and q g p 9
then G is a p-sol¨ able group. Proof. By Feit]Thompson’s Theorem w6x, we may assume 2 g p 9 l p Ž G .. Assume first
HALL SUBGROUPS
509
Let pr s p R r 4 for all r g p . Since G has a solvable Hall p-subgroup, we conclude that G satisfies Ep r. By assumption, G also satisfies Ep Xr . Next let p g pr , q g prX . If q g p , by the solvability of a Hall p-subgroup of G, G satisfies Ep, q . If q f p , by assumption, G also satisfies Ep, q . In any case we conclude that G satisfies Ep, q . Therefore G satisfies the two conditions of Theorem 3 with p s pr . It then follows from Theorem 3 that G is pr-separable for all r g p . This forces G to be p-solvable. The proof of Theorem 5 is complete.
ACKNOWLEDGMENT I would like to thank my Ph.D. supervisor Professor Alexandre Turull. He gave me many good suggestions and much help in writing this paper. Without his help this paper would have never been finished.
REFERENCES 1. Z. Arad, Groups with S2 9-subgroups and S39-subgroups, Israel J. Math. 81 Ž1983., 76]83. 2. Z. Arad and D. Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston J. Math. 7, No. 1 Ž1981., 23]32. 3. Z. Arad and E. Fisman, On finite factorizable groups, J. Algebra 86 Ž1984., 523]548. 4. Z. Arad and M. B. Ward, New criteria for the solvability of finite groups, J. Algebra 77 Ž1982., 234]240. 5. Z. Du, Strengthened Hall’s conjecture for the solvability of finite groups, submitted for publication. 6. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 Ž1963., 775]1029. 7. E. Fisman, On the product of two finite solvable groups, J. Algebra 80 Ž1983., 517]536. 8. D. Gorenstein, ‘‘Finite Groups, I,’’ 2nd ed., Chelsea, New York, 1980. 9. R. M. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 Ž1983., 304]311. 10. P. Hall, On the Sylow systems of a soluble group, Proc. Lond. Math. Soc. Ž 3 . 43 Ž1937., 316]323. 11. P. Hall, Theorems like Sylow’s, Proc. Lond. Math. Soc. Ž 3 . 6 Ž1956., 286]404. 12. M. Herzog, P. Longobardi, M. Maj, and A. Mann, Finite groups containing many subgroups are solvable, to appear. 13. E. I. Spitznagel, Jr., Hall subgroups of certain families of finite groups, Math. Z. 97 Ž1967., 259]290 14. J. S. Williams, Prime graph components of finite groups, J. Algebra 81 Ž1983., 304]311.
JOURNAL OF ALGEBRA ARTICLE NO.
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. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9 ;
Ž2.
G satisfies Ep j q4 and Ep 9j p4 for all p g p , q g p 9.
Theorem. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9 ;
Ž2.
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 to w8x. 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 Hall p 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 n F K ny1 F ??? F K 1 F K 0 s G, where each factor K irK iq1 Ž0 F i F n y 1. 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 G satisfies Ep9 for all p g p Ž G . w11x. Hall also conjectured: G is solvable if and only if G satisfies Ep, q for all p, q g p Ž G . w11x. 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 that the alternating groups A n Ž n G 5. do not contradict the conjecture. Spitznagel, Jr. checked Hall’s Conjecture for many, but not all, of the Chevalley groups w13x. Finally, Arad and Ward w4x proved Hall’s Conjecture by using the classification of finite simple groups. They also proved that if G satisfies E2 9 and E39 , then G must be solvable w4x. Guralnick studied finite simple groups which satisfy Ep9 for some prime p g p Ž G . and showed that such groups are very restricted w9x. Arad and Fisman proved the same result in w3x. In w3, 7x, Arad and Fisman proved that finite simple groups satisfying Ep and Ep 9 are very restricted. They also generalized Hall’s result w11x and proved that a finite group G is solvable if and only if G satisfies E k , for some k such that 2 F k F < G < y 1, where E k means that G has a Hall p1 , p 2 , . . . , pk 4 -subgroup for all p1 , p 2 , . . . , pk 4 : p Ž G ., where p1 , p 2 , . . . , pk are distinct. Arad and Chillag w2x also proved the interesting result that a finite group having a nilpotent Hall p-subgroup with 2 g p is p 0-solvable, where p 0 s p R 24 . Most of the above work concerns the solvability of finite groups. As we know, p-separable Žp-solvable. groups are generalizations of solvable groups. A natural question to ask is whether we can use Hall subgroups to characterize p-separable Žp-solvable. groups. For p-separable groups, the following result is well known: THEOREM w8, Theorem 6.3.5x. Ž1. Ž2. Ž3.
Let G be a p-separable group. Then
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9; G satisfies Ep, q for all p g p , q g p 9.
What about the inverse of the above theorem? Namely, if G satisfies Ž1., Ž2., and Ž3. is G p-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 that Ž1, 2. or Ž1, 3. is sufficient for p-separability of a finite group G. In general, Ž2, 3. is not sufficient since PSLŽ2, 7. with p s 34 provides a counterexample. However, in the case where 2 f p and
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 that Ž1, 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 that Ž2, 3. does imply that G is p-solvable when 2 f p and
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. G is a p-separable group if and only if Ž1.
G satisfies Ep and Ep 9;
Ž2.
G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9.
Proof. Ž«. follows immediately from Hall’s Theorem, e.g. w8, Theorem 6.3.5x. Let us prove the other direction. Ž¥. Suppose Theorem 1 is false and let G be a minimal counterexample. 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 p Ž G .. We divide the proof into several steps. Ž1. G is 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. Thus Ž1. holds. Let A g Hall p Ž G ., B g Hall p 9Ž G .. It is easy to see that G s AB. By Burnside’s p aq b Theorem, either
504
ZHAOWEI DU
implies that M contains a non-trivial proper normal subgroup ² K G : of G. This is contrary to Ž1., and thus Ž2. holds. Ž3. 2 g p 9. Otherwise, 2 g p . Then B is of odd order and solvable w6x. This contradicts Ž2.. Thus Ž3. holds. Ž4.
¦
G
;s¦O
p9
Ž Mi .
Mi B
;s¦O
p9
Ž Mi .
B
;: B z G
implies that B contains a non-trivial proper normal subgroup ² K G : of G. This contradicts Ž1., so that Ž5. holds. Ž6. Final contradiction. Since M j Ž2 F j F r . is of odd order, M j is p-constrained. On the other hand, by w1, Theorem 8.1.2x, G is also p-stable. By Sylow’s Theorem, without loss of generality, we may assume P s A F M j Ž1 F j F r .. According to w8, Theorem 8.1.11x, we have M j s Op9 Ž M j . NM jŽ Z Ž J Ž P . . . ,
for all 2 F j F r .
By Ž5. Op9Ž M j . s 1, hence ZŽ J Ž P ..e } M j for all 2 F j F r. Let X s NG Ž ZŽ J Ž P .... Then < G : X < s 2 a, for some a. Let M1 s PQ for some Q g Syl 2 Ž G .. Then G s XQ. Therefore 1 /² Z Ž J Ž P . . G: s² Z Ž J Ž P . . X Q: s² Z Ž J Ž P . . Q: : M1 z G implies that M1 contains a non-trivial proper normal subgroup ² ZŽ J Ž P .. G : of G. This contradicts Ž1., so that Ž6. holds. The proof of Theorem 1 is complete. THEOREM 2. G is a p-sol¨ able group if and only if Ž1. Ž2.
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j s for all q g p 9 and s : p .
Proof. Ž«. Let G be a p-solvable group, and we prove that G satisfies Ž1. and Ž2.. By Theorem 1, G satisfies Ž1.. G also satisfies Ep j q4.
HALL SUBGROUPS
505
Now all we need to do is to show that G satisfies Ep 9j s . For every s : p , we show that G is p 9 j s-separable. Let K be any chief factor of G. Since G is p-solvable, K is either a p 9-group or a p-group for some p g p . If p g s , then K is p 9 j s-group. If p g p R s , then K is a p 9 j s 4 9-group. Therefore G is p 9 j s-separable and G satisfies Ep 9j s . By Hall’s Theorem, e.g. w8, Theorem 6.3.5x, G satisfies Ep 9j s . Ž¥. By Theorem 1, G is p-separable. Let K be any chief factor of G. Then K is either a p 9-group or a p-group. If K is a p-group, by Ž2., K satisfies Es for all s : p Ž K .. By Hall’s Theorem, e.g. w8, Theorem 6.4.5x, K is a solvable group. This forces K to be a p-group for some p g p . This means that G is p-solvable. Hence the proof of Theorem 2 is complete. COROLLARY 1. G is a p-sol¨ able group if and only if Ž1. Ž2. Ž3.
G satisfies Ep and Ep 9; G satisfies Ep j q4 and Ep 9j p4 for all p g p and q g p 9; G has a Hall p-subgroup which is sol¨ able.
Proof. Corollary 1 follows easily from Theorem 1. When p s p4 , we have the following simple conditions. COROLLARY 2. G is a p-sol¨ able group if and only if Ž1. Ž2.
G satisfies Ep9; G satisfies Ep, q for all q g p Ž G ..
Proof. Corollary 2 is a special case of Theorem 2. Remarks. Ž1. By Feit]Thompson’s Odd Order Paper w6x, when 2 f p , Assumption Ž3. in Corollary 1 can be omitted. Ž2. By a result w5x which says that G is solvable if and only if G satisfies E2, p for all p, when p s 2, Assumption Ž1. in Corollary 2 can be omitted.
2. NEW CHARACTERIZATIONS WITH CFSG In this section, we provide some new characterizations of p-separable and p-solvable groups by using CFSG. THEOREM 3. G is a p-separable group if and only if Ž1. Ž2.
G satisfies Ep and Ep 9; G satisfies Ep, q for all p g p and q g p 9.
Proof. Ž«. follows immediately from Hall’s Theorem, e.g. w8, Theorem 6.3.5x. Let us prove the other direction.
506
ZHAOWEI DU
Ž¥. Suppose Theorem 3 is false and let G be a minimal counterexample. To make our proof symmetric in p and p 9, we assume for convenience that p Ž G . s p j p 9, i.e., p 9 is the complement of p in p Ž G .. We divide the proof into several steps. Ž1. G is 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 p-separable. Therefore G is p-separable, and this is a contradiction. Thus Ž1. holds. Without loss of generality, we assume 2 g p . We let A g Hall p Ž G ., B g Hall p 9Ž G .. It is easy to see that G s AB. Ž2.
A r with r G 5 a prime and A , A ry1.
ŽII. M11 and either A is solvable or A , M10 . ŽIII. M23 and either B is Frobenius of order 11 ? 23 or B is cyclic of order 23 and A , M22 . ŽIV. PSLŽ2, q . where either q g 11, 29, 594 and A , A 5 or 3 - q k 1Ž4. and A is solvable. ŽV. PSLŽ r, q . with r an odd prime such that Ž r, q y 1. s 1 and either G , PSLŽ5, 2. and < B < s 5 ? 31 or A is a maximal parabolic such that PSLŽ r y 1, q . is involved in A. In particular B is either cyclic or Frobenius. This follows from w3, Theorem 1.1x which classifies all finite non-abelian simple groups which can be written as a product of two proper Hall subgroups with relative prime orders. Ž4. A is not solvable. Assume A is solvable. In this case both the Hall p-subgroup and the Hall p 9-subgroup are solvable. Hence G satisfies Er, s for all r, s g p Ž G .. By Hall’s Conjecture, proved by Arad and Ward w4x, G is solvable. This contradicts Ž1. and shows Ž4.. Ž5. G is not in Case ŽI. and G is not in Case ŽII.. This follows directly from Ž2. and Ž4.. Ž6. G is not in Case ŽIII.. Otherwise, by Ž2., we have G , M23 and B is Frobenius of order 11 ? 23. Then, by assumption, M23 satisfies E3, 23 . But, since the Hall 3, 234 subgroups of G are nilpotent, this contradicts the fact that 234 is a simple prime graph component w14x. Thus Ž6. holds. Ž7. G is not in Case ŽIV..
HALL SUBGROUPS
507
Otherwise, by Ž4., G , PSLŽ2, q . where q g 11, 29, 594 and A , A 5 . If q s 11, then we have p 9 s 114 . This contradicts Ž2.. If q s 29, then we have p 9 s 7, 294 . This implies that PSLŽ2, 29. satisfies E3, 29 . But, since the Hall 3, 294 -subgroups of G are nilpotent, this contradicts the fact that 294 is a simple prime graph component w14x. If q s 59, then we have p 9 s 29, 594 . This implies that PSLŽ2, 59. satisfies E5, 59 . But, since the Hall 5, 594 -subgroups of G are nilpotent, this contradicts the fact that 594 is a simple prime graph component w14x. Ž8. G is not in Case ŽV.. Otherwise, assume first G , PSLŽ5, 2. and < B < s 5 ? 31. Then PSLŽ5, 2. satisfies E3, 31 , and therefore GLŽ5, 2. satisfies E3, 31. But this contradicts w13, Corollary 2.1.6x. So we can assume G , PSLŽ r, q . with r an odd prime such that Ž r, q y 1. s 1 and A is a maximal parabolic such that PSLŽ r y 1, q . is involved in A. If q ) 3, let X be a Borel subgroup of G such that X F A, and X s PH, where P is a Sylow p-subgroup of G and H is a Cartan subgroup of G in X. By assumption G satisfies Ep, r for all r g p 9, so there is some Sylow r-subgroup R of G such that PR is a subgroup of G. By w13, Theorem 5.3x, R F X. This implies r g p for all r g p 9 which contradicts Ž2.. So we may assume q F 3. In this case q s 2 or 3. Assume first q s 2, G , PSLŽ r, 2., r G 3 a prime. In this case PSLŽ r, 2. s SLŽ r, 2. s GLŽ r, 2.. Let p g p 9 by assumption, G satisfies E2, p . Reference w13, Theorem 2.3.2x says that p s 3 Žalso r F 5.. This implies
G satisfies Ep and Ep 9;
Ž2.
G satisfies Es j q4 for all s : p and q g p 9 l p Ž G ..
Proof. We only need to prove the ‘‘if’’ part. By Theorem 3, G is p-separable. Let K be any chief factor of G. Then K is either a p-group or a p 9-group. To finish the proof, we only need to consider the case where K is a p-group and prove that K is solvable. For each s : p , there
508
ZHAOWEI DU
exists some q g p 9 l p Ž G .. By Ž2., G satisfies Es j q4. Hence K satisfies Es . So K has Hall s-subgroups for all s : p . It then follows from Hall’s Theorem, e.g. w8, Theorem 6.4.5x, that K is solvable. Therefore the proof is complete. COROLLARY 3. G is a p-sol¨ able group if and only if Ž1. Ž2. Ž3.
G satisfies Ep and Ep 9; G satisfies Ep, q for all p g p and q g p 9; G has a Hall p-subgroup which is sol¨ able.
Proof. We only need to prove the ‘‘if’’ part. By Theorem 3, G is p-solvable. Let K be any chief factor of G. Then K is either a p-group or a p 9-group. To finish the proof, we only need to consider the case that K is a p-group and prove that K is solvable. Let H be a solvable Hall p-subgroup of G. Since K is isomorphic to some section of H, K is solvable as we desired. The proof is complete. Remark. By Feit]Thompson’s Odd Order Paper w6x, when 2 f p , Assumption Ž3. in Corollary 3 can be omitted.
3. FURTHER DISCUSSION From Sections 1 and 2, we know that either Ž1, 2. or Ž1, 3. of Hall’s Theorem, e.g. w8, Theorem 6.3.5x, implies the p-solvability of groups. Therefore we might expect that Ž2, 3. also implies the p-solvability of groups. Unfortunately, it is not so. PSLŽ2, 7. provides a counterexample with p s 34 which satisfies E3, 2 and E3, 7 , but PSLŽ2, 7. is a finite non-abelian simple group. However, we can prove the following theorem: THEOREM 5. Ž1. Ž2.
Assume 2 f p and
Ep j q4 and Ep 9j p4 for all p g p l p Ž G . and q g p 9 l p Ž G .; Ep, q for all p g p and q g p 9
then G is a p-sol¨ able group. Proof. By Feit]Thompson’s Theorem w6x, we may assume 2 g p 9 l p Ž G .. Assume first
HALL SUBGROUPS
509
Let pr s p R r 4 for all r g p . Since G has a solvable Hall p-subgroup, we conclude that G satisfies Ep r. By assumption, G also satisfies Ep Xr . Next let p g pr , q g prX . If q g p , by the solvability of a Hall p-subgroup of G, G satisfies Ep, q . If q f p , by assumption, G also satisfies Ep, q . In any case we conclude that G satisfies Ep, q . Therefore G satisfies the two conditions of Theorem 3 with p s pr . It then follows from Theorem 3 that G is pr-separable for all r g p . This forces G to be p-solvable. The proof of Theorem 5 is complete.
ACKNOWLEDGMENT I would like to thank my Ph.D. supervisor Professor Alexandre Turull. He gave me many good suggestions and much help in writing this paper. Without his help this paper would have never been finished.
REFERENCES 1. Z. Arad, Groups with S2 9-subgroups and S39-subgroups, Israel J. Math. 81 Ž1983., 76]83. 2. Z. Arad and D. Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston J. Math. 7, No. 1 Ž1981., 23]32. 3. Z. Arad and E. Fisman, On finite factorizable groups, J. Algebra 86 Ž1984., 523]548. 4. Z. Arad and M. B. Ward, New criteria for the solvability of finite groups, J. Algebra 77 Ž1982., 234]240. 5. Z. Du, Strengthened Hall’s conjecture for the solvability of finite groups, submitted for publication. 6. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 Ž1963., 775]1029. 7. E. Fisman, On the product of two finite solvable groups, J. Algebra 80 Ž1983., 517]536. 8. D. Gorenstein, ‘‘Finite Groups, I,’’ 2nd ed., Chelsea, New York, 1980. 9. R. M. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 Ž1983., 304]311. 10. P. Hall, On the Sylow systems of a soluble group, Proc. Lond. Math. Soc. Ž 3 . 43 Ž1937., 316]323. 11. P. Hall, Theorems like Sylow’s, Proc. Lond. Math. Soc. Ž 3 . 6 Ž1956., 286]404. 12. M. Herzog, P. Longobardi, M. Maj, and A. Mann, Finite groups containing many subgroups are solvable, to appear. 13. E. I. Spitznagel, Jr., Hall subgroups of certain families of finite groups, Math. Z. 97 Ž1967., 259]290 14. J. S. Williams, Prime graph components of finite groups, J. Algebra 81 Ž1983., 304]311.