On Intersection of Primary Subgroups in the Group Aut(F4(2))

Abstract

Let G be a finite group and A and B be its subgroups. By definition, M is the set of subgroups that are minimal by inclusion among all subgroups of type A ∩ B, g ∈ G, and m consists of those elements of the set M whose order is minimal. Denote by MinG(A,B) (resp. minG(A,B)) the subgroup, generated by the set M (resp. m). First this kind of groups was introduced in [1]. Evidently, minG(A,B) 6 MinG(A,B) and the following three conditions are equivalent: a) A ∩B ̸= 1 for any g ∈ G; b) MinG(A,B) ̸= 1; c) minG(A,B) ̸= 1. If S ∈ Sylp(G) then subgroups minG(S, S) ̸= 1 can be described in many interesting cases. It give us a description of pairs of subgroups (A,B) with the condition minG(A,B) ̸= 1 for primary subgroups and sometimes for nilpotent subgroups A and B. For example, in [2, Theorem 1] it is proved that MinG(A,B) 6 F (G) for any pair of abelian subgroups A and B of G, where F (G) is the Fitting subgroup of G (the greatest normal nilpotent subgroup of G). It was proved in [3] that if G is an almost simple group with socle L2(q), q > 3, and S ∈ Sylp(G), then minG(S, S) = MinG(S, S) = S for the Mersenne prime q = 2 − 1, and the equalities minG(S, S) = MinG(S, S) = 1 hold for all others q, exception q = 9. For q = 9 ∗v1i9z52@mail.ru †nuzhin2008@rambler.ru c ⃝ Siberian Federal University. All rights reserved