Finite groups with an affine map of large order
Abstract
Let $G$ be a group. A function $G\rightarrow G$ of the form $x\mapsto x^{\alpha}g$ for a fixed automorphism $\alpha$ of $G$ and a fixed $g\in G$ is called an affine map of $G$. In this paper, we study finite groups $G$ with an affine map of large order. More precisely, we show that if $G$ admits an affine map of order larger than $\frac{1}{2}G$, then $G$ is solvable of derived length at most $3$. We also show that more generally, for each $\rho\in\left(0,1\right]$, if $G$ admits an affine map of order at least $\rhoG$, then the largest solvable normal subgroup of $G$ has derived length at most $4\lfloor\log_2(\rho^{1})\rfloor+3$.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.10047
 Bibcode:
 2020arXiv200410047B
 Keywords:

 Mathematics  Group Theory;
 Primary: 20F14. Secondary: 20D05;
 20D25;
 20D45
 EPrint:
 19 pages