Khukhro, E.I.
(2008)
*Groups with an automorphism of prime order that is almost regular in the sense of rank.*
Journal of the London Mathematical Society, 77
(1).
pp. 130-148.
ISSN 0024-6107

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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 r be the (PrÃ¼fer) rank of the fixed-point subgroup CG(Ï�). 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 â�¤ slant N â�¤ 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. Â© 2007 London Mathematical Society.

Additional Information: | First published online: December 11, 2007 |
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Keywords: | Group theory |

Subjects: | G Mathematical and Computer Sciences > G100 Mathematics |

Divisions: | College of Science > School of Mathematics and Physics |

Related URLs: | |

ID Code: | 15596 |

Deposited On: | 31 Oct 2014 15:53 |

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