Finite Soluble Groups with Permutable Subnormal Subgroups

Abstract

A finite group G is said to be a PST -group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST -groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT -groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PSTgroups. Our techniques and results provide a unified point of view for T -groups, PT -groups, and PST -groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.

On self-normalizing subgroups of finite groups

“…These classes have been studied in detail, with many characterizations available (see [1], [2], [3], [4], [5], [6], [7], [14], [15]). …”

“…r Proof of Theorem 8. The arguments in the proof of Theorem 13 with Agrawal's result replaced by Zacher's result (see Theorem 1) show that (1) implies (2). It is clear that (2) implies (3) and (4) implies (5).…”

Corrigendum. A note on finite -groups

“…Therefore X satisfies U Ã p , that is, all chief factors of X of order divisible by p are cyclic and isomorphic when regarded as X -modules. Since every subgroup H of N is subnormal and p 0 -perfect, it follows that H permutes with Q by [1]. Hence Q normalizes H and consequently all p 0 -elements of G induce power automorphisms in N. r Proof of Theorem A completed.…”

Infinite groups with many permutable subgroups

“…There are many papers, in which several properties of finite soluble P Tgroups have been considered (see, for example, [1,4,2]). …”