Contemporary Mathematics Hall Subgroups of Finite Groups Danila Olegovitch Revin and Evgenii Petrovitch Vdovin Abstract. In this paper, we complete the classi¯cation of Hall subgroups in ¯nite almost simple groups. We also prove that a ¯nite group satis¯es D¼ if and only if all its composition factors satisfy D¼. 1. Introduction Let ¼ be a set of primes. A ¼-subgroup H of G is called a Hall ¼-subgroup if jG : Hj is not divided by any prime r 2 ¼. Following [25], we say that a ¯nite group G satis¯es E¼ (or just that G is an E¼-group), if G contains a Hall ¼-subgroup; G is a C¼-group if G is an E¼-group and all Hall ¼-subgroups of G are conjugate; and G is a D¼-group if G is a C¼-group and every ¼-subgroup of G is contained in some Hall ¼-subgroup of G. The set of E¼-groups is known to be closed under normal subgroups and factor groups but it is not closed under extensions. Since a normal subgroup of an E¼- group satis¯es E¼, it is important to describe ¯nite simple groups with E¼-property. Moreover, if S is a ¯nite nonabelian simple E¼-group and Aut(S) satis¯es E¼, then by [21, Corollary 3.6], every extension of S by a E¼-group satis¯es E¼. Thus we also consider almost simple E¼-groups. The set of C¼-groups is known to be closed under extensions but it is not closed under normal subgroups. V.A. Vedernikov in [44], using CFSG, proved the following lemma. [44, Lemma 3] Let G satisfy E¼, M · H · G and M E G. If H=M satis¯es E¼, then H also does. It is immediate from this lemma that the set of C¼-groups is closed under factor groups. The main result of [22] states that if 2 62 ¼, then E¼ and C¼ are equivalent. In particular, in this case a ¯nite group G satis¯es E¼ if and only if every composition 2000 Mathematics Subject Classi¯cation. Primary 20D20; Secondary 20D06, 20G40. Key words and phrases. Finite simple groups, Hall subgroups, Groups of Lie type, Algebraic groups, Alternating groups, Normalizer of maximal torus. The work is supported by RFBR, grant N 05-01-00797, Universities of Russia, grant N UR.04.01.202. The second author is partially supported by University of Padua, Italy. °c 0000 (copyright holder) 1 2 D.O.REVIN AND E.P.VDOVIN factor of G satis¯es E¼. In [43], the authors complete the classi¯cation of Hall subgroups of odd order in ¯nite simple groups, so the classi¯cation of all ¯nite E¼-groups with 2 62 ¼ is obtained. The set of D¼-groups is known to be closed under factor groups. L.A. Shemet- kov and V.D. Mazurov, respectively, added the following problems to \Kourovka notebook" [45]: 3.62. Is an extension of a D¼-group by a D¼-group again a D¼-group? 13.33. Is a normal subgroup of a D¼-group again a D¼-group? The result by F. Gross in [22] implies an a±rmative answer to the second problem in case 2 62 ¼. Using the classi¯cation of Hall subgroups of odd order in ¯nite simple groups, the authors in [43] obtain an a±rmative answer to the ¯rst problem in case 2 62 ¼. F. Gross proved the following proposition [23, Proposition 1.1]. Proposition 1.1. Let S be either An(q) or Cn(q). Assume S has a Hall ¼-subgroup with 3 62 ¼. Then all Hall ¼-subgroups of S are conjugate in S and a Hall ¼-subgroup of S has a Sylow tower. Further, if S · G · Aut S, then G has a Hall ¼-subgroup, all Hall ¼-subgroups of G are conjugate in G and a Hall ¼-subgroup of G is solvable. In view of this proposition, F. Gross made the following conjectures. Conjecture 1.2. The above result holds true if S is allowed to be any of 2 2 the classical simple group, i.e., if S is one of An(q), Bn(q), Cn(q), Dn(q), An(q ), 2 2 or Dn(q ). Conjecture 1.3. Let G be a ¯nite group and ¼ a set of primes with 3 62 ¼. Then G has a Hall ¼-subgroup if and only if each composition factor of G has a Hall ¼-subgroup. Conjecture 1.4. If H is a Hall ¼-subgroup of the ¯nite simple group G and 2 3 62 ¼, then either H has a Sylow tower or H = G = B2(q). In this paper, we complete the classi¯cation of Hall subgroups in ¯nite almost simple groups. We prove the three conjectures posed by F.Gross and give a±rmative answers to Problems 3.62 and 13.33 from \The Kourovka Notebook" mentioned above. The authors thank Professor Anatolii S. Kondratiev and Professor Viktor D. Mazurov for helpful discussions. 2. Notations Our notations are standard. For abstract groups, they agree with those in [17].For a group G by Aut G we denote the automorphism group of G and Inn G is a subgroup of Aut G generated by inner automorphisms. A ¯nite group G is called almost simple if there is a ¯nite simple group K with K · G · Aut K. If n is a rational integer, ¼ a set of primes, then by ¼(n) we denote the set of all prime divisors of n, by n¼ we denote the maximal divisor t of n such that ¼(t) ⊆ ¼. For a ¯nite group G, ¼(G) = ¼(jGj) by de¯nition. We say that a Hall ¼-subgroup M of G is nontrivial if M 6= f1g and M is not a Sylow subgroup of G. If n is a positive rational integer, r is an odd prime and (r; n) = 1, then by e(r; n) we denote the minimal positive rational integer e with ne ´ 1(mod r). If n is an odd rational integer, let e(2; n) = 1 if n ´ 1(mod 4) and e(2; n) = 2 if n ´ ¡1(mod 4). HALL SUBGROUPS OF FINITE GROUPS 3 For groups of Lie type and linear algebraic groups, our notation agrees with that in [7] and [26], respectively. Denote by Gsc a universal group of Lie type. Then we call every factor group Gsc=Z, where Z · Z(Gsc), a group of Lie type. Almost in all cases, Gsc=Z(Gsc) is simple and we say that Gad = Gsc=Z(Gsc) is of adjoint type. If G is a universal group of Lie type, then every automorphism of G=Z(G) can be considered as an automorphism of G and vice versa. We shall use this fact below without any reference. 2 2 2 2 2 2 If G is isomorphic to An(q ), Dn(q ), or E6(q ), we say that G is de¯ned 2 3 3 3 over GF (q ), if G ' D4(q ) we say that G is de¯ned over GF (q ) and we say that G is de¯ned over GF (q) for other ¯nite groups of Lie type. The ¯eld GF (q) in all cases is called the base ¯eld of G. If G is a universal ¯nite group of Lie type with the base ¯eld GF (q), then there is a positive integer k(= j©+j in most cases) and a polynomial f(t) 2 Z[t] such that jGj = f(q) ¢ qk and (q; f(q)) = 1 (see [7, Theorems 9.4.10 and 14.3.1]). We denote this polynomial by fG(t). If G is not universal, then there is a universal group K with G = K=Z and we de¯ne fG(t) = fK (t). Assume that G is a connected simple linear algebraic group de¯ned over an algebraically closed ¯eld of a positive characteristic p. Let σ be an endomorphism of G such that G = CG(σ) is a ¯nite set. Then σ is said to be a Frobenius map and p0 G = O (G) appears to be a ¯nite group of Lie type. Moreover, all ¯nite groups of Lie type both split and twisted can be derived in this way. By [39, Theorem 30], p0 every automorphism of G = O (G) can be extended to an automorphism of G. Let M · Aut G and for every x 2 M, ¯x an extension x 2 Aut G. If x 2 M \ G (clearly G · Aut G), then x = x 2 G is the conjugation by x. We say that a subgroup R of G is invariant under M · Aut G (or just M-invariant), if for any x 2 M; we have that R is invariant under its (¯xed before) extension x 2 Aut G. In particular, if M · G, we have that M · NG(R). If G is simply connected, p0 then G = G = O (G) and if G is of adjoint type, then G = Gb is the group of inner-diagonal automorphisms of G. If R is a σ-stable reductive subgroup of G, then R \ G = R \ G is called a reductive subgroup of G. If R is of maximal rank, then R \ G is also of maximal rank. Note that if R is σ-stable of maximal rank, then R = T ¢G1 ¤:::¤Gk, where T is some σ-stable maximal torus£ of R¤and G1;:::; Gk are subsystem subgroups of G. Furthermore, G1 ¤ ::: ¤ Gk = R; R . It is known that R = T G1 ¤ ::: ¤ Gm. We call subgroups G1;:::;Gm subsystem subgroups of G. In general, m · k and, for all ®i i, the base ¯eld of Gi is equal to GF (q ), where ®i ¸ 1.