A note on groups with countable automorphism groups (Q1068938)

The subject of this paper is the analysis of the behaviour of the elements of finite order of a group G when countability conditions are imposed on the automorphism group Aut G of G. The main result is the following Theorem. Let G be a finite-by-nilpotent group such that the Hirsch-Plotkin radical \((=\) the maximum normal locally nilpotent subgroup) of Aut G is countable; then the Sylow subgroups of G have finite exponents, and G has finite exponent, if it is a torsion group. A stronger result is obtained when the group of all central automorphisms of G is countable. Examples show that these results are not true if the countability condition is imposed on the Baer radical of Aut G.