Power maps in finite groups

Matt Larson

Published 2017-07-20Version 1

In recent work, Pomerance and Shparlinski have obtained results on the number of cycles in the functional graph of the map $x \mapsto x^a$ in $\mathbb{F}_p^*$. We prove similar results for other families of finite groups. In particular, we obtain estimates for the normal order, average order, and the extremal number order of the number of cycles for $(\mathbb{Z}/n\mathbb{Z})^*$ and cyclic groups. We find a lower bound for the number of cycles for symmetric groups, and we obtain expressions for the number of cycles for dihedral groups and $SL_2(\mathbb{F}_q)$. We also show that the cyclic group of order $n$ minimizes the number of cycles among all nilpotent groups of order $n$. Finally, we pose several open problems.