On a class of non-solvable groups

Abstract

In this paper, we use the properties of subgroups with given order to study the structure of finite groups. The main result is as follows:Let G be a group and P be a Sylow p-subgroup of G. Suppose that 1<d≤|P|. If every subgroup H of P with |H|=d is M-supplemented in G, then every non-abelian pd-G-chief factor A/B satisfies one of the following conditions:(1) A/B≅PSL(2,7) and p=7; A/B≅PSL(2,11) and p=11;(2) A/B≅PSL(2,2t) and p=2t+1>3 is a Fermat prime;(3) A/B≅PSL(n,q), n≥3 is a prime, (n,q−1)=1 and p=qn−1/q−1;(4) A/B≅M11 and p=11; A/B≅M23 and p=23;(5) A/B≅Ap and p≥5.