On the Norm of the Abelian p-Group-Residuals

Let G be a group. Dp(G)=⋂H≤GNG(H′(p)) is defined and, the properties of Dp(G) are investigated. It is proved that Dp(G)=P[A], where P=D(P) is the Sylow p-subgroup and A=N(A) is a Hall p′-subgroup of Dp(G), respectively. Furthermore, it is proved in a group G that (1) Dp(G)=1 if and only if CG(G′(p))=1; (2) Op′(Dp(G))≤Z∞(Op(G)) and (3) if Z(G′(p))=1, then CG(G′(p))=Dp(G).