Criterions of Supersolubility for Products of Supersoluble Groups

Abstract

Let H and T be subgroups of a group G. Then we call H conditionally permutable (or in brevity, c-permutable) with T in G if there exists an element x ∈ G such that HT x = T xH. If H is c-permutable with T in < H, T >, then we call H completely cpermutable with T in G. By using the above concepts, we will give some new criterions for the supersolubility of a finite group G = AB, where A and B are both supersoluble groups. In particular, we prove that a finite group G is supersoluble if and only if G = AB, where both A, B are nilpotent subgroups of the group G and B is completely c-permutable in G with every term in some chief series of A. We will also give some applications of our new criterions.