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

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