Radical locally finite T-groups

Summary

Introduction

A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation in G. The study of this class of groups begins with the publication of a paper of Dedekind in 1896. He characterizes the finite groups in which every subgroup is normal. These groups, called Dedekind groups, are obvious examples of T-groups. The extension of Dedekind's result to infinite groups was proved by Baer in 1933.

Theorem (Dedekind, Baer)All the subgroups of a group G are normal if and only if G is abelian or the direct product of a quaternion group of order 8, an elementary abelian 2-group and an abelian group with all its elements of odd order.

In 1942, E. Best and O. Taussky [5] prove that every finite group with cyclic Sylow subgroups is a T-group. Later G. Zacher characterized soluble finite Tgroups by means of Sylow towers properties (see [12]). However, the decisive result about the structure of T-groups in the finite soluble universe was obtained by Gaschütz in 1957 ([8]).

Theorem (Gaschütz)Let G be a finite soluble group. Then G is a T-group if and only if it has an abelian normal Hall subgroup L of odd order such that G/L is a Dedekind group and the elements of G induce power automorphisms in L.