Let G be a finite group.Let D={(g,g)|g∈C},the main diagonalsubgroup of G x G.In this paper,we consider the suitable generalized normalities orindex of D in G x G,some interesting results are obtained.
Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.
