James C. Beidleman

Some characterizations of finite groups in which semipermutability is a transitive relation

Abstract Let G be a finite group and let H be a subgroup of G. H is said to be semipermutable in G if H permutes with every subgroup K of G with (|H|, |K|) = 1. A number of new characterizations of finite solvable BT-groups are given, where a BT-group is one in which semipermutability is a transitive relation.

A Local Approach to Certain Classes of Finite Groups

Abstract We develop several local approaches for the three classes of finite groups: T-groups (normality is a transitive relation) and PT-groups (permutability is a transitive relation) and PST-groups (S-permutability is a transitive relation). Here a subgroup of a finite group G is S-permutable if it permutes with all the Sylow subgroup of G.

Subnormal, permutable, and embedded subgroups in finite groups

The purpose of this paper is to study the subgroup embedding properties of S-semipermutability, semipermutability, and seminormality. Here we say H is S-semipermutable (resp. semipermutable) in a group Gif H permutes which each Sylow subgroup (resp. subgroup) of G whose order is relatively prime to that of H. We say H is seminormal in a group G if H is normalized by subgroups of G whose order is relatively prime to that of H. In particular, we establish that a seminormal p-subgroup is subnormal. We also establish that the solvable groups in which S-permutability is a transitive relation are precisely the groups in which the subnormal subgroups are all S-semipermutable. Local characterizations of this result are also established.

On the fitting core of a formation

In this note we shall be concerned with an aspect of formations of solvable groups, especially of saturated formations. If a class of groups is a formation, it is closed with respect to forming quotient groups and subdirect products. Very quickly one is confronted with the limitation of this concept: in particular, the class of supersolvable groups — an example that comes to mind easily — is not closed with respect to normal products, one of the two criteria for being a Fitting class. The Fitting core V of a class C defines the subclass of groups that do not lead out of the class. In other words, the class V is denned

On PC-hypercentral and CC-hypercentral groups

This paper investigates the effects of the imposition of some chain conditions on groups having a generalized central series. It also provides a characterization of PC-groups and CC-groups with finite abelian section rank

ON A CLASS OF SUPERSOLUBLE GROUPS

A subgroup \(H\) of a finite group \(G\) is said to be S-semipermutable in \(G\) if \(H\) permutes with every Sylow \(q\)-subgroup of \(G\) for primes \(q\) not dividing \(|H\)|. A finite group \(G\) is an MS-group if the maximal subgroups of all the Sylow subgroups of \(G\) are S-semipermutable in \(G\). The aim of the present paper is to characterise the finite MS-groups. DOI: 10.1017/S0004972714000306

On hypercentral factor groups from certain classes

Throughout the rest of this paper, the hypercenter, nilpotent residual, Fitting subgroup and Frattini subgroup of G will be denoted by Z ðGÞ, g ðGÞ, FitðGÞ and FðGÞ respectively. In [18] the first two authors introduce a new class of groups, the socalled T1-groups. A group G is a T1-group if G=Z ðGÞ is a T-group. Some of the basic properties of T1-groups are established in [18]. The present work is a continuation of [18]. In the light of [18, Theorem A] and Theorem 1, we begin with

On pairwise mutually permutable products

Abstract Some results about products of pairwise mutually permutable subgroups are presented in this paper. It is shown that this kind of products behaves well with respect to some well-known classes of groups. For instance, we show that all factors have only simple chief factors if the product has this property. This is necessary but not sufficient: we need that the factors belong to the subclass of PST-groups to make sure that the product has only simple chief factors (see Theorems 5 and 6).

On the Structure of the Normal Subgroups of a Group: Nilpotency

A group G is said to have property v if, whenever N is a normal non-nilpotent subgroup of G, there is a finite non-nilpotent G-quotient of N. Polycyclic-by-finite groups, free groups, subgroups of GL(«, Z), subgroups of finitely generated abelian-by-nilpotent-by-finite groups, and free metanilpotent groups satisfy property v. Fit (G) is the Fitting subgroup of G, φ (G) is the Frattini-subgroup of G, and φ{ (G) is the intersection of all maximal subgroups of G of finite index in G (here φ{ (G) = G if no maximal subgroups of finite index in G exist). A group G has property v if and only if φ{ (G) and Fit (G) are nilpotent subgroups of G and Fit (G/0f (G)) = Fit(G)/f (G). A group G of finite rank has property v if and only if G is soluble-by-finite, φ (G) and Fit (G) are nilpotent subgroups of G, and Fit(G/^(G)) = 1980 Mathematics Subject Classification (1985 Revision): 20E26, 20F18, 20E28.

Injectors of locally soluble FC-groups

For a Fitting setF of a locally soluble FC-group, the existence and local conjugacy ofF is established. In particular, the locally nilpotent injectors are described. Normally embedded subgroups of locally soluble FC-groups are characterized in terms of Fischer sets.