Hall subgroups and the pronormality

1 1 Sobolev Institute of Mathematics, Novosibirsk, Russia revin@math.nsc.ru Novosibirsk, November 14, 2013

Definition A subgroup H of a group G is said to be pronormal if H and H g are conjugate in H, H g for every g G. The notation H prn G means H is a pronormal subgroup of G. Examples of pronormal subgroups Normal subgroups; Maximal subgroups; Sylow subgroups in finite groups; Sylow subgroups of normal subgroups in finite groups.

Definition Let p P. A subgroup P of a group G is called a Sylow p-subgroup if P is a power of p (i. e., P is a p-group) while G : P is not divisible by p. Theorem (L. Sylow, 1872) Let G be a finite group and let p be a prime. Then E p C p D p G possesses a Sylow p-subgroup; every two Sylow p-subgroups are conjugate; every p-subgroup of G is included in a Sylow p-subgroup of G. We denote by Syl p (G) the set of Sylow p-subgroup of a finite group G.

Corollary Let G be a finite group. Then P prn G for every P Syl p (G). Corollary Let G be a finite group and A G. Then P prn G for every P Syl p (A). Corollary (Frattini Argument) Let G be a finite group, A G, and P Syl p (A). Then G = AN G (P).

The Frattini Argument is closely connected with the pronormality. If A G and H A then G = AN G (H) iff H and H g are conjugate in A for every g G (we write H A = H G, were H G = {H g g G}). As a consequence, if H prn G then G = AN G (H).

A natural generalization of the concept of a Sylow p-subgroup is that of a π-hall subgroup. We fix a set π P. Put π = P \ π. Definition A subgroup H of a finite group G is called a π-hall subgroup if every prime divisor of H belongs to π (H is a π-subgroup) and every prime divisor of G : H belongs to π. The set of all π-hall subgroups of G is denoted by Hall π (G). If π = {p} then Hall π (G) = Syl p (G).

What properties of Sylow subgroups hold for Hall subgroups? What can one say about the pronormality of Hall subgroups?

Hall subgroups have some nice properties. Proposition Let G be a finite group, A G, and H Hall π (G). Then H A Hall π (A) and HA/A Hall π (G/A).

The analogue of Sylow s Theorem fails for Hall subgroups: In Alt 5 of order 60 = 2 2 3 5, there are no elements and subgroups of order 15, hence Alt 5 does not have {3, 5}-Hall subgroups. In GL 3 (2) of order 168 = 2 3 3 7, there are exactly two conjugacy classes of subgroups of order 2 3 3 (= {2, 3}-Hall subgroups): the stabilizers of lines and planes, respectively. Every subgroup of order 12 = 2 2 3 (= a {2, 3}-Hall subgroup) of Alt 5 is a point stabilizer, and all point stabilizers are conjugate. On the other hand, Alt 5 includes a {2, 3}-subgroup (123), (12)(45) Sym 3 which acts without fixed points.

Theorem (P. Hall, 1928) Let G be a solvable finite group and let π be a set of primes. Then E π C π G possesses a π-hall subgroup; every two π-hall subgroups of G are conjugate; D π every π-subgroup of G is included in a π-hall subgroup of G. Corollary Let G be a solvable group and H Hall π (G). Then H prn G.

Definition (P.Hall) Given a set of primes π, we say that a finite group G satisfies E π C π D π if Hall π (G) (i.e., there exists a π-hall subgroup in G); if G satisfies E π and every two π-hall subgroups of G are conjugate; if G satisfies C π and every π-subgroup of G is included in a π-hall subgroup. A group G satisfying E π (resp., C π, D π ) is called an E π - (resp., C π -, D π -) group. Given a set of primes π, we also denote by E π, C π, and D π the classes of all finite E π -, C π -, and D π - groups, respectively.

There exist sets π P such that E π = C π. Theorem (F.Gross, 1987) If 2 π then E π = C π. Proposition If E π = C π for some π then π-hall subgroups of finite groups are pronormal. If H Hall π (G) then, given g G, we have H, H g E π = C π, and hence H and H g are conjugate in H, H g. Corollary Hall subgroups of odd order are pronormal.

Let π P and E π C π. Take X E π \ C π and non-conjugate U, V Hall π (X ). Let n π. Consider Y = X X X X }{{} n times and τ Aut(Y ), where τ : (x 1, x 2,..., x n 1, x n ) (x 2, x 3,..., x n, x 1 ) for all x 1, x 2,..., x n 1, x n X. Let G = Y τ X Z n. The subgroups H = V U } {{ U U }, n 1 times K = U } U {{ U } V n 1 times of Y are π-hall in Y and in G. Moreover, H τ = K. But H and K are not conjugate in Y, and hence are non-pronormal in G.

In the talk we are concerned with the following problems: What can one say about E π -groups in which there exists a pronormal π-hall subgroup? (the results are obtained in collaboration with Prof. E.P.Vdovin) What can one say about E π -groups in which every π-hall subgroup is pronormal? (the results are obtained in collaboration with Prof. Wenbin Guo)

On the existence of pronormal subgroups in E π -groups in collaboration with Prof. E.P.Vdovin

Theorem (E.Vdovin, D.R., 2013) In C π -groups, every π-hall subgroup is pronormal. This theorem is equivalent to the following statement. Theorem (E.Vdovin, D.R., 2013) Let G C π, H Hall π (G), and H M G. Then M C π. In order to prove these theorems, the classification of finite simple groups and the classification of Hall subgroups in such groups are used.

Main ingredients of our proof: The theorem on the number of classes of conjugate π-hall subgroups in finite simple groups (the Class Number Theorem); The converse to Gross theorem on the existence of π-hall subgroups; The theorem on the pronormality of Hall subgroups of finite simple groups.

The same ingredients are sufficient to prove a statement stronger than the pronormality of π-hall subgroups in C π -groups. Theorem 1 (E.Vdovin, D.R., new) If G E π then G has a pronormal π-hall subgroup. Thus, the class of groups containing a pronormal π-hall subgroup coincides with E π. Theorem 2 (E.Vdovin, D.R., new) If G E π and A G then there exists H Hall π (A) such that H prn G. Theorem 3 (Frattini Argument, E.Vdovin, D.R., new) If G E π and A G then there exists H Hall π (A) such that G = AN G (H).

Theorem 3 (Frattini Argument, E.Vdovin, D.R., new) If G E π and A G then G = AN G (H) for some H Hall π (A). Corollary (E.Vdovin, D.R., 2006) If G D π and A G then A D π. Corollary (E.Vdovin, D.R., 2010) If G C π, H Hall π (G), and A G then HA C π. Corollary (E.Vdovin, D.R., 2011) If G E π and A G then Hall π (G/A) = {HA/A H Hall π (G)}.

Corollary (E.Vdovin, D.R., new) Let A G. Then G E π iff A E π, G/A E π, and there is H Hall π (A) such that H A = H G. Corollary (E.Vdovin, D.R., new) Let G E π, A Aut(G) and ( G, A ) = 1. Then there exists an A-invariant H Hall π (G).

Theorem 1 (E.Vdovin, D.R., new) If G E π then G has a pronormal π-hall subgroup. Corollary (E.Vdovin, D.R., 2013) In C π -groups, every π-hall subgroup is pronormal. Corollary (E.Vdovin, D.R., 2013) Let G C π, H Hall π (G), and H M G. Then M C π.

Theorem 1 If G E π then H prn G for some H Hall π (G). Theorem 2 If G E π and A G then H prn G for some H Hall π (A). Theorem 3 (Frattini Argument for Hall subgroups) If G E π and A G then G = AN G (H) for some H Hall π (A). Compare these statements with their analogs for Sylow subgroups.

Corollary 1 to Sylow s Theorem If G is a group then P prn G for every P Syl p (G). Corollary 2 to Sylow s Theorem If G is a group and A G then P prn G for every P Syl p (A). Corollary 3 to Sylow s Theorem (Frattini Argument for Sylow subgroups) If G is a group and A G then G =AN G (P) for every P Syl p (A).

Thus, in Theorems 1 3, one cannot replace for some with for every.

Theorem 1 If G E π then H prn G for some H Hall π (G). Theorem 2 If G E π and A G then H prn G for some H Hall π (A). Theorem 3 (Frattini Argument for Hall subgroups) If G E π and A G then G = AN G (H) for some H Hall π (A). Can one replace the condition G E π in Theorems 2 and 3 with the weaker one A E π?

Let π = {2, 3} and let A = GL 3 (2). Then A has exactly two classes of conjugate ( π-hall subgroups ) with ( the representatives ) GL 2 (2) 1 U = and V =. 1 GL 2 (2) The first one consists of the line stabilizers in the natural representation of A, and the second one consists of the plane stabilizers. Consider the automorphism ι : x (x t ) 1, x A, of A (here x t is the transpose of x). Then ι = 2 and ι interchanges U A and V A. Let G = A ι. Then K A K G for every K Hall π (A). In particular, K is non-pronormal in G and G AN G (K). Suppose, H Hall π (G). Then G = HA and G stabilizes (H A) A. But (H A) A coincides either with U A or with V A, and hence G does not stabilize (H A) A. A contradiction. Thus, G / E π.

Thus, in Theorems 2 and 3, one cannot replace G E π with A E π.

In our proof of this Theorem, we use the classification of finite simple groups and the classification of Hall subgroups in such groups. If G is a finite group then we denote by k π (G) the number of classes of conjugate π-hall subgroups of G. Class Number Theorem (E.Vdovin and D.R., 2010, mod CFSG) Let S E π be a finite simple group. Then the following statements hold: if 2 π, then k π (S) = 1; if 3 π, then k π (S) {1, 2}; if 2, 3 π, then k π (S) {1, 2, 3, 4, 9}. In particular, k π (S) is a bounded π-number.

Furthermore, if S G then we denote by k G π (S) the number of classes of conjugate π-hall subgroups K of S such that K = H S for some H Hall π (G). As a consequence of the Class Number Theorem, we have CNT-Corollary Let G E π possess a unique minimal normal subgroup S and let S be a finite simple group. Then the following statements hold: if 2 π, then k G π (S) = 1; if 3 π, then k G π (S) {1, 2}; if 2, 3 π, then k G π (S) {1, 2, 3, 4, 9}. In particular, k G π (S) is a π-number.

If G E π and A G then H G = H A for some H Hall π (A). Firstly, let A be the unique minimal normal subgroup of G and let A be simple. Denote by Ω the set of classes of conjugate π-hall subgroups K of S such that K = H A for some H Hall π (G). Then Ω = k G π (A) and G acts on Ω in a natural way. Consider H Hall π (G) and K = (H A) A = {H a A a A} Ω. Since H A H, H leaves invariant K, and hence H G K. Thus, K G divides G : H and K G is a π -number. On the other hand, K G Ω = k G π (A). In view of CNT-Corollary, one of the following statements holds: K G is a π-number; 2, 3 π, k G π (A) = 9, and K G {5, 7}. In the first case, K G = 1, i. e. (H A) A = (H A) G. In the second case, the power of every orbit L G of G on Ω that differs from K G is at most 4. Hence L G is a π-number and L G = 1.

Now consider the situation where A is a minimal normal subgroup ( A is not necessarily simple and G does not necessarily have a unique minimal normal subgroup).

Definition Suppose A, B, H are subgroups of G such that B A. Then N H (A/B) = N H (A) N H (B) is the normalizer of A/B in H. If x N H (A/B) then x induces an automorphism of A/B by Ba Bx 1 ax. Thus there exists a homomorphism N H (A/B) Aut(A/B). The image of N H (A/B) under this homomorphism is denoted by Aut H (A/B) and is called the group of H-induced automorphisms of A/B. Theorem (F.Gross, 1986) Let 1 = G 0 < G 1 <... < G n = G be a composition series for a finite group G which is a refinement of a chief series for G. If Aut G (G i /G i 1 ) E π for all i = 1,..., n, then G E π. Theorem (E.Vdovin and D.R., 2011) Let 1 = G 0 < G 1 <... < G n = G be a composition series for a finite group G. If G E π then Aut G (G i /G i 1 ) E π for all i.

In the considered situation, A = S 1 S n, where S i are simple and S i = S g i 1 for some g i G, i = 1,..., n. Then g 1,..., g n is a right transversal for N G (S 1 ) in G. By the converse to Gross Theorem, Aut G (S 1 ) = N G (S 1 )/C G (S 1 ) E π. By the above-considered case, S 1 = Inn(S1 ) includes an Aut G (S 1 )-invariant conjugacy class K of π-hall subgroups. Let U K. Then U g i Hall π (S i ), V = U g i i = 1,..., n Hall π (A), and V G = V A, i.e., G = AN G (V ).

Now, one can investigate the general situation in Theorem 3 by using induction on G.

Theorem 1: In an E π -group G, H prn G for some H Hall π (G). Theorem (E.Vdovin, D.R., 2012) In the finite simple groups, the Hall subgroups are pronormal. Thus, we can assume that G is not simple. Let A be a minimal normal subgroup of G. Then A = S 1 S n, where S i are simple. Moreover, K Hall π (A) K prn A. By Theorem 1, G = AN G (K) for some K Hall π (A). By induction, H/K prn N G (K)/K for some H/K Hall π (N G (K)/K) and H Hall π (G). H/K prn N G (K)/K HA/A prn AN G (K)/A = G/A. Proposition Let H Hall π (G), A G, and HA/H prn G/A; H A prn A; (H A) A = (H A) G. Then H prn G.

Conjecture 1 Every Hall subgroup is pronormal in its normal closure. If H Hall π (G) then H G = O π (G). Conjecture 1 If H Hall π (G) and H G = G then H prn G. A subgroup H of G is said to be strongly pronormal if K g is conjugate to a subgroup of H in H, K g for every g G and K H. Conjecture 2 Every pronormal Hall subgroup is strongly pronormal. Conjecture 3 Every E π -group contains a strongly pronormal π-hall subgroup.

On E π -groups in which every π-hall subgroup is pronormal In collaboration with Prof. Guo Wenbin

By analogy with P.Hall s notation, we will say that a group G satisfies P π (is a P π -group, belongs to the class P π ) if G E π and every π-hall subgroups in G is pronormal. Theorem (1) If G E π is simple then G P π. (2) C π P π E π. (3) If C π E π then C π P π E π.

sx= {G G is isomorphic to a subgroup of H X}; qx= {G G is an epimorphic image of H X}; s n X= {G G is isomorphic to a subnormal subgroup of H X} r 0 X= {G N i G (i = 1,..., m) with G/N i X and m N i = 1}; i=1 n 0 X= {G N i G (i = 1,..., m) with N i X and G = N 1,..., N m }; ex= {G G possesses a series 1 = G 0 G 1 G m = G with G i /G i 1 X (i = 1,..., m)}. If π = {3, 5} then the classes D π, C π, P π, and E π are not s-closed: SL 2 (16) D π while SL 2 (4) = Alt 5 / E π.

Table: Is it true that cx = X, X {D π, E π, C π, P π }? c D π E π C π P π s no no no no q yes yes yes yes s n yes yes no no r 0 yes yes yes yes n 0 yes no yes no e yes no yes no Theorem (Wenbin Guo, D.R., mod CFSG) The following statements hold: (A) cp π = P π for every set π of primes and c {q, r 0 }. (B) If c {s, s n, n 0, e}, then cp π P π for a set π of primes. (C) If c {s n, e} and cp π = P π for a set π of primes then ce π = E π and cc π = C π.

Corollary (Wenbin Guo, D.R., mod CFSG) Let c {s, q, s n, r 0, n 0, e}. Then the following statements are equivalent: (1) cp π = P π for every set π of primes; (2) ce π = E π and cc π = C π for every set π of primes. Corollary (Wenbin Guo, D.R., mod CFSG) For a set π of primes, P π is a saturated formation. Corollary (Wenbin Guo, D.R.) There exist sets π (for example, π = {2, 3}) of primes such that P π is not a Fitting class. Proposition (mod CFSG) If E π = C π and c {q, s n, r 0, n 0, e} then cp π = P π. In particular, P π is a Fitting class.

Thank you for your attention!