Hermann Heineken

Groups with subnormality for all subgroups that are not finitely generated

SummaryWe characterize the groups given in the title in the case of locally finite, locally nilpotent and radical groups.SuntoI gruppi con tutti i sottogruppi nonfinitamente generati subnormali sono caratterizzati nella classe dei gruppi localmente finiti, localmente nilpotenti e radicali.

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.

The subnormal embedding of complete groups

In [ 1) Dark records a result of Philip Hall that the symmetric group S, of degree three cannot be embedded subnormally in a finite perfect group, that is, a group which is equal to its own derived subgroup. Now S, is the smallest complete group, that is, a group with trivial centre and no outer automorphisms and our objective here is to show that Hall’s result is a specific case of a more general phenomenon. In fact we have

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 a class of locally finite T-groups

Abstract Radical locally finite groups with min-p for all primes p in which every descendant subgroup is normal are studied in the paper. It turns out that these groups are precisely T-groups, that is, groups whose subnormal subgroups are normal.

The Structure of Mutually Permutable Products of Finite Nilpotent Groups

We consider mutually permutable products G = AB of two nilpotent groups. The structure of the Sylow p-subgroups of its nilpotent residual is described.

Pairwise -connected Products of Certain Classes of Finite Groups

Abstract Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.

On Sylow permutable subgroups of finite groups

Abstract A subgroup H of a group G is called Sylow permutable, or S-permutable, in G if H permutes with all Sylow p-subgroups of G for all primes p. A group G is said to be a PST-group if Sylow permutability is a transitive relation in G. We show that a group G which is factorised by a normal subgroup and a subnormal PST-subgroup of odd order is supersoluble. As a consequence, the normal closure S G {S^{G}} of a subnormal PST-subgroup S of odd order of a group G is supersoluble, and the subgroup generated by subnormal PST-subgroups of G of odd order is supersoluble as well.