On The Norm Of The Abelian P-Group-Residuals

Let G be a group. D-p(G) = boolean AND(H <= G) N-G(H'(p)) is defined and, the properties of D-p(G) are investigated. It is proved that D-p(G) = P [A], where P = D(P) is the Sylow p-subgroup and A = N(A) is a Hall p'-subgroup of D-p(G), respectively. Furthermore,it is proved in a group G that (1) D-p(G) = 1 if and only if C-G(G'(p)) = 1; (2) O-p'(D-p(G)) <= Z(infinity)(O-p(G)) and (3) if Z(G'(p)) = 1, then C-G(G'(p)) = D-p(G).