1. Noncoprime action of a cyclic group
- Author
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Ercan, Gülin and Güloğlu, İsmail Ş.
- Subjects
Mathematics - Group Theory - Abstract
Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of $A$ counted with multiplicities. In the present paper we consider the case $A$ is cyclic and obtain that the nilpotent length of $G$ is at most $2\ell(A)$ if $|G|$ is odd. More generally we prove that the nilpotent length of $G$ is at most $2\ell(A)+ \mathbf{c}(G;A)$ when $G$ is of odd order and $A$ normalizes a Sylow system of $G$ where $\mathbf{c}(G;A)$ denotes the number of trivial $A$-modules appearing in an $A$-composition series of $G$., Comment: The proof of Theorem 2.6 is incorrect. Without this theorem the main claim of the paper becomes unproven
- Published
- 2023