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 \(C_G(A)/(C_G(A))'\) divides \(|G:C_G(A)|\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Blackburn, Automorphisms of finite p-groups, J. Algebra, 3 (1966), 28–29.
I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society (Providence, RI, 2008).
I. M. Isaacs and G. Navarro, Normal p-complements and fixed elements, Arch. Math. (Basel), 95 (2010), 207–211.
M. Y. Kızmaz, A short note on Isaacs-Navarro’s theorem, Arch. Math. (Basel), 113 (2019), 225–227.
The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.11.1 (2021), http://www.gap-system.org.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kizmaz, M.Y. On the rank and exponent of the fixed points of coprime actions. Acta Math. Hungar. 166, 107–114 (2022). https://doi.org/10.1007/s10474-021-01198-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-021-01198-8