Abstract

Let � be an automorphism of prime order p of a finite group G, and let CG ( � ) be its fixed-point subgroup. When � is regular, that is, CG ( � ) = 1, the group G is nilpotent by Thompson's theorem. The "almost regular" results of Fong and Hartley-Meixner-Pettet were giving the existence of a nilpotent subgroup of index bounded in terms of p and | CG ( � ) |. We prove the rank analogues of these results, when "almost regular" in the hypothesis is interpreted as a restriction on the rank r of CG ( � ), and the conclusion is sought as nilpotency modulo certain bits of bounded rank. The classification is used to prove almost solubility in the coprime case: the rank of G / S ( G ) is bounded in terms of r and p. For soluble groups the Hall-Higman-type theorems are combined with the theory of powerful q-groups to obtain almost nilpotency, even without the coprimeness condition: there are characteristic subgroups R less-than or slanted equal to N less-than or slanted equal to G such that N / R is nilpotent and the ranks of R and G / N are bounded in terms of r and p. Examples show that our results are in a sense best-possible. © 2006 Elsevier Inc. All rights reserved.