Groups with an automorphism of prime order that is almost regular in the sense of rank

Let φ be an automorphism of prime order p of a finite group G , and let r be the (Prüfer) rank of the fixed‐point subgroup C G (φ). It is proved that if G is nilpotent, then there exists a characteristic subgroup C of nilpotency class bounded in terms of p such that the rank of G / C is bounded in terms of p and r . For infinite (locally) nilpotent groups a similar result holds if the group is torsion‐free (due to Makarenko), or periodic, or finitely generated; but examples show that these additional conditions cannot be dropped, even for nilpotent groups. As a corollary, when G is an arbitrary finite group, the combination with the recent theorems of the author and Mazurov gives characteristic subgroups R ⩽ slant N ⩽ slant G such that N / R is nilpotent of class bounded in terms of p while the ranks of R and G / N are bounded in terms of p and r (under the additional unavoidable assumption that p ∤ | G | if G is insoluble); in general it is impossible to get rid of the subgroup R . The inverse limit argument yields corresponding consequences for locally finite groups.