Makarenko, N. Yu. and Khukhro, E. I.
(2006)
*Finite groups with an almost regular automorphism of order four.*
Algebra and Logic, 45
(5).
pp. 326-343.
ISSN 0002-5232

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Item Type: | Article |
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Item Status: | Live Archive |

## Abstract

P. Shumyatsky's question 11.126 in the "Kourovka Notebook" is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism Ï� of order 4 having exactly m fixed points, then G has a normal series G â�¥ H â�¥ N such that |G/H| â�¤ f(m), the quotient group H/N is nilpotent of class â�¤ 2, and the subgroup N is nilpotent of class â�¤ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes KovÃ¡cs' theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors' previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class â�¤ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. Â© 2006 Springer Science+Business Media, Inc.

Additional Information: | Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006 |
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Keywords: | Logic, Algebra |

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

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

Related URLs: | |

ID Code: | 15708 |

Deposited On: | 11 Nov 2014 18:18 |

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