The permutizer of the main diagonal subgroups in direct products

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.