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 Ï� be an automorphism of prime order p of a finite group G, and let CG ( Ï� ) be its fixed-point subgroup. When Ï� is regular, that is, CG ( Ï� ) = 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 ( Ï� ) |. 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 ( Ï� ), 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. Â© 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 |

Related URLs: | |

ID Code: | 15709 |

Deposited On: | 14 Nov 2014 13:23 |

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