Finite groups with an automorphism of prime order whose centralizer has small rank

Khukhro, E. I. and Mazurov, V. D. (2006) Finite groups with an automorphism of prime order whose centralizer has small rank. Journal of Algebra, 301 (2). pp. 474-492. ISSN 0021-8693

Full content URL: http://dx.doi.org/10.1016/j.jalgebra.2006.02.039

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Item Type:Article
Item Status:Live Archive

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.

Keywords:Algebra, Groups
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:15709
Deposited On:14 Nov 2014 13:23

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