Abstract

Let � be an automorphism of prime order p of a finite group G, and let r be the (Prüfer) rank of the fixed-point subgroup CG(�). 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. © 2007 London Mathematical Society.