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 |
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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 |
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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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