Introduction Hall subgroups in finite simple groups HALL SUBGROUPS IN FINITE SIMPLE GROUPS Evgeny P. Vdovin1 1Sobolev Institute of Mathematics SB RAS Groups St Andrews 2009 Introduction Hall subgroups in finite simple groups The term “group” always means a finite group. By π we always denote a set of primes, π0 is its complement in the set of all primes. A rational integer n is called a π-number, if all its prime divisors are in π, by π(n) we denote all prime divisors of a rational integer n. For a group G we set π(G) to be equal to π(jGj) and G is a π-group if jGj is a π-number. A subgroup H of G is called a π-Hall subgroup if π(H) ⊆ π and π(jG : Hj) ⊆ π0. A set of all π-Hall subgroups of G we denote by Hallπ(G) (note that this set may be empty). According to P. Hall we say that G satisfies Eπ (or briefly G 2 Eπ), if G possesses a π-Hall subgroup. If G 2 Eπ and every two π-Hall subgroups are conjugate, then we say that G satisfies Cπ (G 2 Cπ). If G 2 Cπ and each π-subgroup of G is included in a π-Hall subgroup of G, then we say that G satisfies Dπ (G 2 Dπ). The number of classes of conjugate π-Hall subgroups of G we denote by kπ(G). Introduction Hall subgroups in finite simple groups The term “group” always means a finite group. By π we always denote a set of primes, π0 is its complement in the set of all primes. A rational integer n is called a π-number, if all its prime divisors are in π, by π(n) we denote all prime divisors of a rational integer n. For a group G we set π(G) to be equal to π(jGj) and G is a π-group if jGj is a π-number. A subgroup H of G is called a π-Hall subgroup if π(H) ⊆ π and π(jG : Hj) ⊆ π0. A set of all π-Hall subgroups of G we denote by Hallπ(G) (note that this set may be empty). According to P. Hall we say that G satisfies Eπ (or briefly G 2 Eπ), if G possesses a π-Hall subgroup. If G 2 Eπ and every two π-Hall subgroups are conjugate, then we say that G satisfies Cπ (G 2 Cπ). If G 2 Cπ and each π-subgroup of G is included in a π-Hall subgroup of G, then we say that G satisfies Dπ (G 2 Dπ). The number of classes of conjugate π-Hall subgroups of G we denote by kπ(G). Introduction Hall subgroups in finite simple groups The term “group” always means a finite group. By π we always denote a set of primes, π0 is its complement in the set of all primes. A rational integer n is called a π-number, if all its prime divisors are in π, by π(n) we denote all prime divisors of a rational integer n. For a group G we set π(G) to be equal to π(jGj) and G is a π-group if jGj is a π-number. A subgroup H of G is called a π-Hall subgroup if π(H) ⊆ π and π(jG : Hj) ⊆ π0. A set of all π-Hall subgroups of G we denote by Hallπ(G) (note that this set may be empty). According to P. Hall we say that G satisfies Eπ (or briefly G 2 Eπ), if G possesses a π-Hall subgroup. If G 2 Eπ and every two π-Hall subgroups are conjugate, then we say that G satisfies Cπ (G 2 Cπ). If G 2 Cπ and each π-subgroup of G is included in a π-Hall subgroup of G, then we say that G satisfies Dπ (G 2 Dπ). The number of classes of conjugate π-Hall subgroups of G we denote by kπ(G). Introduction Hall subgroups in finite simple groups Elementary properties of Hall subgroups 1 If A E G and H 2 Hallπ(G), then HA=A 2 Hallπ(G=A) and H \ A 2 Hallπ(A). 2 Assume that G possesses a subnormal series feg = G0 < G1 < : : : < Gk−1 < Gk = G such that the order of each section either is a π-number, or is divisible by at most one prime from π. Then G satisfies Dπ, i. e., G possesses a π-Hall subgroup H and each π-subgroup of G is conjugate to a subgroup of H. In such case we say that a π-Hall subgroup of G is standard, otherwise a π-Hall subgroup of G is called nonstandard. Introduction Hall subgroups in finite simple groups Elementary properties of Hall subgroups 1 If A E G and H 2 Hallπ(G), then HA=A 2 Hallπ(G=A) and H \ A 2 Hallπ(A). 2 Assume that G possesses a subnormal series feg = G0 < G1 < : : : < Gk−1 < Gk = G such that the order of each section either is a π-number, or is divisible by at most one prime from π. Then G satisfies Dπ, i. e., G possesses a π-Hall subgroup H and each π-subgroup of G is conjugate to a subgroup of H. In such case we say that a π-Hall subgroup of G is standard, otherwise a π-Hall subgroup of G is called nonstandard. Introduction Hall subgroups in finite simple groups Extension Lemma If A E G, π(G=A) ⊆ π, and M 2 Hallπ(A), then there exists H 2 Hallπ(G) with H \ A = M if and only if G, acting by conjugation, leaves invariant the set fMa j a 2 Ag. Let π = f2; 3g, G = GL3(2) = SL3(2) be a group of order 168 = 23 · 3 · 7. Then G has exactly two classes of π-Hall subgroups with representatives ! ! GL (2) ∗ 1 ∗ 2 and : 0 1 0 GL2(2) The map ι : x 2 G 7! (xt )−1 is an automorphism of order 2 of G. It interchanges classes of π-Hall subgroups, hence the group Gb = G : hιi does not possess a π-Hall subgroup. Introduction Hall subgroups in finite simple groups Extension Lemma If A E G, π(G=A) ⊆ π, and M 2 Hallπ(A), then there exists H 2 Hallπ(G) with H \ A = M if and only if G, acting by conjugation, leaves invariant the set fMa j a 2 Ag. Let π = f2; 3g, G = GL3(2) = SL3(2) be a group of order 168 = 23 · 3 · 7. Then G has exactly two classes of π-Hall subgroups with representatives ! ! GL (2) ∗ 1 ∗ 2 and : 0 1 0 GL2(2) The map ι : x 2 G 7! (xt )−1 is an automorphism of order 2 of G. It interchanges classes of π-Hall subgroups, hence the group Gb = G : hιi does not possess a π-Hall subgroup. Introduction Hall subgroups in finite simple groups Extension Lemma If A E G, π(G=A) ⊆ π, and M 2 Hallπ(A), then there exists H 2 Hallπ(G) with H \ A = M if and only if G, acting by conjugation, leaves invariant the set fMa j a 2 Ag. Let π = f2; 3g, G = GL3(2) = SL3(2) be a group of order 168 = 23 · 3 · 7. Then G has exactly two classes of π-Hall subgroups with representatives ! ! GL (2) ∗ 1 ∗ 2 and : 0 1 0 GL2(2) The map ι : x 2 G 7! (xt )−1 is an automorphism of order 2 of G. It interchanges classes of π-Hall subgroups, hence the group Gb = G : hιi does not possess a π-Hall subgroup. Introduction Hall subgroups in finite simple groups Theorem (Alternating and symmetric groups) Let π be a set of primes. 1 All possibilities for Symn to contain a nonstandard π-Hall subgroup are listed in table below. 2 The following statements are equivalent: Symn 2 Cπ; Symn 2 Eπ; Altn 2 Eπ; Altn 2 Cπ. 3 M 2 Hallπ(Altn) if and only if there exists M0 2 Hallπ(Symn) such that M = M0 \ Altn. n π \ π(Symn) H 2 Hallπ(Symn) prime π((n − 1)!) Symn−1 7 f2; 3g Sym3 × Sym4 8 f2; 3g Sym4 o Sym2 Introduction Hall subgroups in finite simple groups Theorem (Alternating and symmetric groups) Let π be a set of primes. 1 All possibilities for Symn to contain a nonstandard π-Hall subgroup are listed in table below. 2 The following statements are equivalent: Symn 2 Cπ; Symn 2 Eπ; Altn 2 Eπ; Altn 2 Cπ. 3 M 2 Hallπ(Altn) if and only if there exists M0 2 Hallπ(Symn) such that M = M0 \ Altn. n π \ π(Symn) H 2 Hallπ(Symn) prime π((n − 1)!) Symn−1 7 f2; 3g Sym3 × Sym4 8 f2; 3g Sym4 o Sym2 Introduction Hall subgroups in finite simple groups Sporadic Eπ-groups, 2 62 π G π \ π(G) G π \ π(G) G π \ π(G) M11 f5; 11g M12 f5; 11g M22 f5; 11g Ru f7; 29g M24 f5; 11g M23 f5; 11g f11; 23g f11; 23g 0 Fi23 f11; 23g Fi24 f11; 23g Ly f11; 67g 0 J1 f3; 5g J4 f5; 7g O N f3; 5g f3; 7g f5; 11g f5; 11g f3; 19g f5; 31g f5; 31g f5; 11g f7; 29g f7; 43g Co1 f11; 23g Co2 f11; 23g Co3 f11; 23g B f11; 23g M f23; 47g f23; 47g f29; 59g Introduction Hall subgroups in finite simple groups π-Hall subgroups of the sporadic groups, 2 2 π G π Structure of H 2 M11 f2; 3g 3 : Q8 : 2 f2; 3; 5g Alt6 : 2 4 M22 f2; 3; 5g 2 : Alt6 4 M23 f2; 3g 2 :(3 × Alt4): 2 4 f2; 3; 5g 2 : Alt6 4 f2; 3; 5g 2 :(3 × Alt5): 2 f2; 3; 5; 7g L3(4): 22 4 f2; 3; 5; 7g 2 : Alt7 f2; 3; 5; 7; 11g M22 6 · M24 f2; 3; 5g 2 : 3 Sym6 J1 f2; 3g 2 × Alt4 f2; 7g 23 : 7 f2; 3; 5g 2 × Alt5 f2; 3; 7g 23 : 7 : 3 11 6 · J4 f2; 3; 5g 2 :(2 : 3 Sym6) Introduction Hall subgroups in finite simple groups Theorem (Revin, 1999) Let G be a finite group of Lie type over a field of characteristic p 2 π.