The intersection map of subgroups and certain classes of finite groups

Abstract

The goals of this paper are twofold. One is to look at the behavior of the collections of permutable subgroups and S-permutable subgroups under the intersection map into a fixed subgroup of a group. The other is to locally analyze the intersection map in connection with $${\mathcal{T}}$$ -, $${\mathcal{PT}}$$ -, and $${\mathcal{PST}}$$ -groups. In particular, we generalize Theorem 1 of Bauman [Arch. Math. (Basel) 25:337–340, 1974] to $${\mathcal{PT}}$$ - and $${\mathcal{PST}}$$ -groups.