The influence of \(IC\bar{s}\)-subgroups on the structure of finite groups

Abstract

A subgroup H of a group G is said to be an \(IC\bar{s}\)-subgroup of G if the intersection of H and [H,G] is contained in H_\(\bar{s}\)G, where H_\(\bar{s}\)G is the maximal s-semipermutable subgroup of G contained in H. Our main result here is the following. Let Ꞙ be a solubly saturated formation containing Մ and E be a normal subgroup of a group G such that G/E ∈ Ꞙ. Let X = E or X = F∗(E). If every non-trivial Sylow subgroup P of X has a subgroup D with 1 < |D| < |P| such that every subgroup of P with order |D| and 4 (if |D| = 2 and P is a non-abelian 2-group) is an \(IC\bar{s}\)-subgroup of G, then G ∈ Ꞙ.