On the rank and exponent of the fixed points of coprime actions

Date
2022-02
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Source Title
Acta Mathematica Hungarica
Print ISSN
0236-5294
Electronic ISSN
Publisher
Springer
Volume
166
Issue
1
Pages
107 - 114
Language
English
Type
Article
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Abstract

Let A be a group acting on a p-group P coprimely. We show that if A centralizes some specified abelian subgroups of P, then A acts trivially on P. As a consequence of this, we obtain that the special rank of CP(A) is strictly less than that of P unless the action of A on P is trivial. Secondly, we prove that if A acts on a group G coprimely and [G,A]=G, then the exponent of CG(A)/(CG(A))′ divides |G:CG(A)|.

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Keywords
Coprime action, Fixed point, Special rank
Citation
Published Version (Please cite this version)