Groups and Lie rings admitting an almost regular automorphism on prime order

It is proved that if a Lie ring $L$ admits an automorphism of prime order $p$ with a finite number $m$ of fixed points and with $pL=L$, then $L$ has a nilpotent subring of index bounded in terms of $p$ and $m$ and whose nilpotency class is bounded in terms of $p$. It is also shown that if a nilpotent periodic group admits an automorphism of prime order $p$ which has a finite number $m$ of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of $m$ and $p$ and whose class is bounded in terms of $p$ (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman–Kreknin–Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.