The permutizer of the main diagonal subgroups in direct products

Abstract

Let G be a finite group, $H\le G$. The permutizer of H in G is defined to be $P_G(H)=\langle x\in G|~H\langle x\rangle =\langle x\rangle H\rangle$. Let $D=\{(g, g)|~g\in G\}$, the main diagonal subgroup of $G\times G$. In this paper, we use the permutizer of D in $G\times G$ to characterize the structure of G, and the following main result is obtained. Main Theorem: Let G be a group, $D=\{(g, g)|~g\in G\}$. Then the group $G\times G$ has a chain of subgroups from D to $G\times G$ with each contained in the permutizer of the previous subgroup if and only if all chief factors T of G have prime order or order 4 with $G/{C_G(T)}\cong S_3$. Finally, we also present two theorems deciding the supersolubility of finite groups.