Finite groups with non-nilpotent maximal subgroups

In this paper, the structure of a finite group under group theoretic restrictions on its non-nilpotent subgroups has been investigated.     

Finite groups with abelian second maximal subgroups

Abstract

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian.     

Finite groups with subnormal second or third maximal subgroups

In the paper, the precise structure of finite groups all of whose second (or all of whose third) maximal subgroups are subnormal is established.     

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.     

Volume invariant and maximal representations of discrete subgroups of Lie groups

Let $\Gamma $ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma $ into $G$ , which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this volume invariant exactly characterizes discrete, faithful representations of $\Gamma $ into $G$ .     

Finite groups in which every 3-Maximal subgroup commutes with all maximal subgroups

The structure of finite groups in which every 3-maximal subgroup commutes with all maximal subgroups is described.     

Topological groups with compact proper invariant subgroups

A construction is given which lets one construct examples of locally compact inductively pronilpotent groups for which all proper invariant subgroups are compact and which do not contain proper open invariant subgroups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 708–710, May, 1990.     

On finite groups with given maximal subgroups

We prove that a finite group whose every maximal subgroup is simple or nilpotent is a Schmidt group. A group whose every maximal subgroup is simple or supersoluble can be nonsoluble, and in this case we prove that its chief series has the form 1 ? K ? G, K }~ PSL ₂(p) for a suitable prime p, |G: K| ≤ 2.     

Finite p-groups with complemented maximal cyclic subgroups

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 780–785, June, 1992.     

Finite primary groups,p ≠ 2, with invariant not completely splittable subgroups

An investigation of finite primary groups, p 2, in which every not completely splittable subgroup is invariant, was carried out.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 551–560, May, 1971.     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.