A survey on power maps in groups

TL;DR

This survey reviews the recent developments in the study of power maps on groups, focusing on the map x↦x^M and summarizing key results and open questions in the field.

Contribution

It compiles and discusses various known results on power maps in groups and presents open questions for future research.

Findings

Collected diverse results on power maps in groups

Identified key open questions in the field

Summarized recent progress and challenges

Abstract

The study of word maps on groups has been of deep interest in recent years. This survey focuses on the case of power maps on groups; $viz.$ the map $x\mapsto x^M$ for a group $G$, and an integer $M\geq 2$. Here, we accumulate various results on the subject and pose some questions.