On a problem of B. Hartley about a small centralizer in finite and locally finite groups

It is proved that if a finite group $G$ has an automorphism of order $n$ with $m$ fixed points, then $G$ has a soluble subgroup whose index and Fitting height are bounded in terms of $m$ and $n$. As a corollary, a problem of B. Hartley is solved in the affirmative: if a locally finite group $G$ has an element with finite centralizer, then $G$ has a subgroup of finite index which has a finite normal series with locally nilpotent factors.