Wielandts program on the study of X -maximal subgroups of finite - PowerPoint PPT Presentation

Wielandt’s program on the study of X -maximal subgroups of finite groups Danila Revin 1 Sobolev Institute of Mathematics, Novosibirsk, Russia revin@math.nsc.ru Yichang, G2D2, August 18, 2019

The report is based on some old ideas expressed by Helmut Wielandt in his talk to the famous Santa-Cruz conference on finite groups in 1979 and in his lectures given in T¨ ubingen in 1963-1964. H. Wielandt, Zusammengesetzte Gruppen: H¨ older Programm heute, The Santa Cruz conf. on finite groups, Santa Cruz, 1979. Proc. Sympos. Pure Math., 37, Providence RI: Amer. Math. Soc., 1980, 161–173. H. Wielandt, Zusammengesetzte Gruppen endlicher Ordnung, Vorlesung an der Universit¨ at T¨ ubingen im Wintersemester 1963/64. Helmut Wielandt: Mathematical Works, Vol. 1, Group theory (ed. B. Huppert and H. Schneider, de Gruyter, Berlin, 1994), 607–516.

Notation and terminology

◮ In the talk, ’a group’ means ’a finite group’ and G is a group. ◮ Always a group G acts on a set Ω on the right: Ω × G → Ω ( ω, g ) �→ ω g ◮ Recall, a series G = G 0 ≥ G 1 ≥ · · · ≥ G n = 1 is said to be normal (resp., subnormal , composition ), if, for all i = 1 , . . . , n , • G i � G ; • G i � G i − 1 (such G i is called subnormal in G , G i �� G ); and • G i ⊳ G i − 1 and G i − 1 / G i is simple. In this case, the sections G i − 1 / G i are called composition factors of G . ◮ Every finite group has a composition series. Up to isomorphism, the multiset of composition factors of G does not depend on the choose of a composition series and is an invariant of G .

◮ We fix a complete class X of finite groups. According to Wielandt, a non-empty class X is said to be complete if X is closed under taking • subgroups ( G ∈ X and H ≤ G ⇒ H ∈ X ), • homomorphic images ( G ∈ X and φ : G → G ∗ ⇒ G φ ∈ X ) and • extensions ( A � G and A , G / A ∈ X ⇒ G ∈ X ). ◮ If G ∈ X then we say that G is an X -group. ◮ Examples of complete classes: • The class G of finite groups is complete. • The class S of finite solvable groups is complete. • Take a subset π of the set P of all primes. A group G is a π -group , if p | | G | ⇒ p ∈ π for all p ∈ P . The classes G π and S π of finite π -groups and finite solvable π -groups, respectively, are complete. • Given n > 0, the class G n of all groups G such that | S | < n for every nonabelian composition factor S of G is complete. ◮ For every complete X , we have S π ⊆ X ⊆ G π , where π = π ( X ) is the set of primes p such that p divides the order of some G ∈ X .

I. X -Maximality vs maximality

A group is an active object and is acting on various sets. A source of actions of G is G itself: G acts • on the set of its elements via (right) multiplication, conjugation etc., • on the set of cosets of a subgroup via multiplication, • on the set of subgroups of G via conjugation etc. If G acts on a set then G acts transitively on every orbit. The group itself is the source of all its transitive actions : every transitive action of G is equivalent to the action of G on the cosets of a point stabilizer. Often the study of transitive actions can be reduced to the study of so-called primitive actions.

In group-theoretical terms, a transitive action of G is primitive if and only if a point stabilizer is a maximal subgroup of G . A subgroup H of G is said to be maximal if H < G and H ≤ K ≤ G implies either K = H or K = G for all subgroups K of G . In other words, for subgroups, the maximality means the maximality among the proper subgroups w.r.t. inclusion. The O’Nan–Scott theorem provides some classification of the faithful primitive actions. An important case of a primitive permutation group is a so-called almost simple group. Recall, a group is said to be almost simple , if it is isomorphic to a group G such that S ∼ = Inn ( S ) ≤ G ≤ Aut ( S ) for a nonabelian simple group S which is the socle of G . Every such G can be considered as a primitive permutation group in which a point stabiliser is a maximal subgroup of G not containing S . There are no hope to obtain the complete list of maximal subgroups of the simple groups.

◮ A subgroup H of G is said to be maximal if H < G and H ≤ K < G ⇒ K = H for all K . In other words, for subgroups, the maximality means the maximality among the proper subgroups w.r.t. inclusion. ◮ For a complete class X , a subgroup H of G is called an X -maximal subgroup or a maximal X -subgroup if H ∈ X and H ≤ K ∈ X ⇒ K = H for all K ≤ G . In other words, for subgroups, the X -maximality means the maximality among the X -subgroups w.r.t. inclusion. Denote by m X ( G ) the set of X -maximal subgroups of G . ◮ If G ∈ X then m X ( G ) = { G } while H < G for any maximal H . ◮ Let p be a prime. Then m G p ( G ) = Syl p ( G ) . ◮ If G is a p -group and H � = 1 is a maximal subgroup of G then there are no complete classes X such that H ∈ m X ( G ) . The definitions are similar, but the maximality and the X -maximality are absolutely distinct concepts.

Let G be an almost simple group such that S = Inn ( S ) ≤ G ≤ Aut ( S ) for nonabelian simple S . Denote X = G | S | , the (complete) class of groups whose n.-a. composition factors are of order less than | S | . Then H is a maximal subgroup of G not including S (i. e. H is a point stabilizer in G considered as a primitive permutation group) iff H ∈ m X ( G ) . Thus, in the almost simple groups, the maximality of a subgroup means an X -maximality for some complete X .

We are interested in the following Main Problem Given a group G and a complete class X , determine m X ( G ) . As well as the study of maximal subgroup is the crucial point in studying the subgroup structure of a group, studying the X -subgroups of a group (for example, a classical problem dating back to works by Galois and Jordan: determine solvable subgroups of the symmetric groups) can be reduced to the above problem.

II. Reduction to factors of a subnormal series: X -submaximal subgroups

Main Problem Given a group G and a complete class X , determine m X ( G ) . Following to an approach practiced in the group theory, it is natural to try to reduced this problem to the factors of a (sub)normal (for example, composition) series of G by replacing X -maximal subgroups with their projections.

Let a series of subgroups G = G 0 ≥ G 1 ≥ . . . ≥ G n − 1 ≥ G n = 1 (1) of a group G be subnormal, i. e. G i � G i − 1 for every i = 1 , . . . , n . For a subgroup H of G we denote by H i = ( H ∩ G i − 1 ) G i / G i the projection of H on the factor G i = G i − 1 / G i . Example: G = G 0 � G 1 � G 2 = 1 (this series is necessarily normal) H 1 = HG 1 / G 1 is the image of H in G / G 1 under G → G / G 1 and H 2 = H ∩ G 1 is the intersection of H and G 1 . Note, that in general, it may happen that: ◮ for given subgroups H i of the factor G i of (1), there is no subgroups of G with projections H i , ◮ H i = K i for subgroups H , K ≤ G and all i = 1 , . . . , n , but H and K are nonisomorphic.

Take a subnormal series G = G 0 ≥ G 1 ≥ . . . ≥ G n − 1 ≥ G n = 1 of a group G (i. e. G i � G i − 1 for every i = 1 , . . . , n ) and a subgroup H ≤ G . If H i are X -maximal in G i = G i − 1 / G i for every i = 1 , . . . , n , then H ∈ m X ( G ) . What about the converse statement?

Take a subnormal series G = G 0 ≥ G 1 ≥ . . . ≥ G n − 1 ≥ G n = 1 of a group G (i. e. G i � G i − 1 for every i = 1 , . . . , n ) and a subgroup H ≤ G . If H i are X -maximal in G i = G i − 1 / G i for every i = 1 , . . . , n , then H ∈ m X ( G ) . What about the converse statement?

Example 1 Let p be a prime and X = G p the class of p -group. Let H ∈ m X ( G ) , i.e., H is a Sylow p -subgroup of G , and A � G . We have a normal series G � A � 1 , HA / A is a Sylow p -subgroup of G / A , i.e. H 1 = HA / A ∈ m X ( G / A ) = m X G 1 � � , H ∩ A is a Sylow p -subgroup of A , i.e. H 2 = H ∩ A ∈ m X ( A ) = m X G 2 � � .

Example 2 Let X = G 168 (or X = S , or X = G { 2 , 3 } ), let G = PGL 2 ( 7 ) , and let A = PSL 2 ( 7 ) ∼ = PGL 3 ( 2 ) . We have a normal series G � A � 1 , where G / A is of order 2 and A = PSL 2 ( 7 ) is simple. | A | = 168 = 2 3 · 3 · 7 , | G | = 2 4 · 3 · 7. A Sylow 2-subgroup H of G is maximal in G . In particular, H ∈ m X ( G ) . But H 2 = H ∩ A / ∈ m X ( A ) .

Wielandt’s idea was to find a generalization of the concept of an X -maximal subgroup which would be close to this one, informative (in particular, differ from the concept of an X -subgroup) and would well behave in the case of intersections with normal (equivalently, with subnormal) subgroups. Definition (Wielandt, 1979) A subgroup H of a group G is called a submaximal X -subgroup (or X -submaximal subgroup, notation H ∈ sm X ( G ) ) if there is a monomorphism φ : G → G ∗ into a finite group G ∗ such that G φ �� G ∗ and H φ = K ∩ G φ for some K ∈ m X ( G ∗ ) . Briefly, H ∈ sm X ( G ) if there are G ∗ and K ∈ m X ( G ∗ ) such that G �� G ∗ and H = K ∩ G . If H ∈ sm X ( G ) and A �� G then H ∩ A ∈ sm X ( A ) . m X ( G ) ⊆ sm X ( G ) . Example of an X -submaximal but not X -maximal subgroup: X := G { 2 , 3 } is the class of { 2 , 3 } -groups; G := PSL 2 ( 7 ) = PGL 3 ( 2 ) ; a Sylow 2-subgroup of G is X -submaximal but not X -maximal.