Rank and order of a finite group admitting a Frobenius-like group of automorphisms

Ercan, G. and Guloglu, I. and Khukhro, Evgeny (2014) Rank and order of a finite group admitting a Frobenius-like group of automorphisms. Algebra and Logic, 53 (3). pp. 258-265. ISSN 0002-5232

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Rank and order of a finite group admitting a Frobenius-like group of automorphisms

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Abstract

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with certain additional restrictions (which are satisfied, in particular, if either |FH| is odd or |H| = 2). In the case where G is a finite p-group such that G = [G, F] it is proved that the rank of G is bounded above in terms of |H| and the rank of the fixed-point subgroup C G (H), and that |G| is bounded above in terms of |H| and |C G (H)|. As a corollary, in the case where G is an arbitrary finite group estimates are obtained of the form |G| ≤|C G (F)| · f(|H|, |C G (H)|) for the order, and r(G) ≤ r(C G (F)) + g(|H|, r(C G (H))) for the rank, where f and g are some functions of two variables.

Additional Information:Translated from Algebra i Logika, Vol. 53, No. 3, pp. 401-412, May-June, 2014.
Keywords:Automorphism, Finite group, Frobenius group, rank, order, JCNotOpen
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:16200
Deposited On:05 Dec 2014 09:57

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