Power maps in finite groups

TL;DR

This paper extends the analysis of cycle structures in power maps from finite fields to various finite groups, providing estimates and identifying minimal cycle counts in certain classes.

Contribution

It generalizes previous results to cyclic, symmetric, dihedral, and special linear groups, and identifies minimal cycle counts in nilpotent groups of fixed order and exponent.

Findings

Cycle estimates for multiple finite groups

Cyclic groups minimize cycles among nilpotent groups of fixed order

New problems posed for future research

Abstract

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 number of cycles for cyclic groups, symmetric groups, 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 $n$ for a fixed exponent. Finally, we pose several problems.