On periodic radical groups in which permutability is a transitive relation

Abstract

A group G is said to be a PT-group if permutability is a transitive relation in the set of all subgroups of G. Our purpose in this paper is to study PT-groups in the class of periodic radical groups satisfying min-p for all primes p.

Semipermutability in Generalised Soluble groups

“…2]. Infinite soluble T-groups and PT-groups were studied in [16] and [5,13,14]. Locally finite PST-groups were studied in [4,19].…”

Groups with Chernikov conjugacy classes in which Sylow permutability is a transitive relation

“…Zacher determined the structure of finite soluble P T -groups in a similar way to Gaschütz' characterization of finite soluble T -groups [10]. Very recently, nice characterizations of some classes of infinite P T -groups have been obtained [7,6].…”

Infinite groups with many permutable subgroups

“…Therefore it is natural to consider the opposite situation, that is the groups whose ascendant subgroups are permutable. A group G is said to be an AP -group if every ascendant subgroup of G is permutable in G. These groups are very close to the groups in which the relation to be a permutable subgroup is transitive (studied in [3] for a class of periodic radical groups). A group G is said to be a P T -group if permutability is a transitive relation in G, that is, if K is a permutable subgroup of H and H is a permutable subgroup of G, then K is a permutable subgroup of G. We note that, if G is an AP -group, then clearly G is a P T -group, because the relation to be an ascendant subgroup is transitive.…”