Let G beagroup. Thesubgroup H of G iscalledself-centralizing,ifthecentralizerH in G, CG(H),iscontained in H. Clearly, an abelian subgroup A of G is self-centralizing if and only if CG(A) = A. We note that if G is abelian, then every subgroup of G is self-centralizing. Furthermore, the trivial subgroup of G is self-centralizing if and only if G is trivial. If H is a subgroup of G, the it is easy to see that the normalizer of H in G, NG(H) is self-centralizing.
