Existence criterion for Hall subgroups of finite groups

In the paper we obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series.Comment: We made some editor corrections in the tex

Finite groups whose maximal subgroups have the hall property

We study the structure of finite groups whosemaximal subgroups have the Hall property. We prove that such a group G has at most one non-Abelian composition factor, the solvable radical S(G) admits a Sylow series, the action of G on sections of this series is irreducible, the series is invariant with respect to this action, and the quotient group G/S(G) is either trivial or isomorphic to PSL2(7), PSL2(11), or PSL5(2). As a corollary, we show that every maximal subgroup of G is complemented. © 2013 Allerton Press, Inc

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and Hg are conjugate in 〈H,Hg〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSLn(q), PSUn(q), E6(q), 2E6(q), where in all cases q is odd and n is not a power of 2, and P Sp2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp2n(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups. The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2m or 2m(22k+1), this group has a nonpronormal subgroup of odd index. If n = 2m, then we show that all subgroups of P Sp2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp2n(q) is still open when n = 2m(22k + 1) and q ≡ ±3 (mod 8). © 2017, Pleiades Publishing, Ltd

On the pronormality of subgroups of odd index in finite simple groups

We prove the pronormality of subgroups of finite index for many classes of simple groups. © 2015, Pleiades Publishing, Ltd

Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in 〈 H, H g〉 for every g ∈ G. In this paper, we determine the finite simple groups of type E 6 (q) and E 6 2 (q) in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.This work was supported by the Russian Science Foundation (project 19-71-10067)