Some new characterizations of $PST$-groups

Abstract

Let H and B be subgroups of a finite group G such that G = N G (H)B. Then we say that H is quasipermutable (respectively S-quasipermutable) in G provided H permutes with B and with every subgroup (respectively with every Sylow subgroup) A of B such that (|H|, |A|) = 1. In this paper we analyze the influence of S-quasipermutable and quasipermutable subgroups on the structure of G. As an application, we give new characterizations of soluble P ST -groups. IntroductionThroughout this paper, all groups are finite and G always denotes a finite group. Moreover p is always supposed to be a prime and π is a subset of the set P of all primes; π(G) denotes the set of all primes dividing |G|.A subgroup H of G is said to be quasinormal or permutable in G if H permutes with every subgroup A of G, that is, HA = AH; H is said to be S-permutable in G if H permutes with every Sylow subgroup of G.As well as T -groups, P T -groups and P ST -groups possess many interesting properties (see Chapter 2 in [1]). The general description of P T -groups and P ST -groups were firstly obtained by Zacher [2] and Agrawal [3], for the soluble case, and by Robinson in [4], for the general case. Nevertheless,