Finite factorizable groups with solvable ℙ2-subnormal subgroups

Abstract

A subgroup H of a finite group G is called ℙ2-subnormal whenever there exists a subgroup chain H = H0 ≤ H1 ≤ ... ≤ Hn = G such that |Hi+1: Hi| divides prime squares for all i. We study a finite group G = AB on assuming that A and B are solvable subgroups and the indices of subgroups in the chains joining A and B with the group divide prime squares. In particular, we prove that a group of this type is solvable without using the classification of finite simple groups.