Finite groups of bounded rank with an almost regular automorphism of prime order

We prove that if a finite group $G$ of rank $r$ admits an automorphism $\varphi$ of prime order having exactly m fixed points, then $G$ has a $\varphi$-invariant subgroup of $(r,m)$-bounded index which is nilpotent of $r$-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite $p$-groups. For reduction to Lie rings powerful $p$-groups are also used. For them a useful fact is proved which allows us to “glue together” nilpotency classes of factors of certain normal series (Theorem 2).