On fixed points of central automorphisms of finite-by-nilpotent groups

The central kernel K(G)K(G) of a group G is the subgroup consisting of all elements fixed by every central automorphism of G. It is proved here that if G is a finite-by-nilpotent group whose central kernel has finite index, then G is finite over the centre, and the elements of finite order of G form a finite subgroup; in particular G is finite, provided that it is periodic. Moreover, if G   is a periodic finite-by-nilpotent group and G/K(G)G/K(G) is a Černikov group, it turns out that G itself is a Černikov group.