Finite groups with an almost regular automorphism of order four

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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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 � of order 4 having exactly m fixed points, then G has a normal series G � H � N such that |G/H| � f(m), the quotient group H/N is nilpotent of class � 2, and the subgroup N is nilpotent of class � 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 Kovács' 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 � 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. © 2006 Springer Science+Business Media, Inc.

Additional Information:Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006
Keywords:Logic, Algebra
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:15708
Deposited On:11 Nov 2014 18:18

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