On the Central Kernel of a Group

Abstract

The central kernel $K(G)$ of a group G is the (characteristic) subgroup consisting of all elements $x\in G$ such that $x^{\gamma }=x$ for every central automorphism $\gamma $ of G. We prove that if G is a finite-by-nilpotent group whose central kernel has finite index, then the full automorphism group $Aut(G)$ of G is finite. Some applications of this result are given.