Word maps with constants on symmetric groups

TL;DR

This paper investigates the behavior of word maps with constants on symmetric groups, revealing limitations of identities and properties of images, and characterizing self-maps as word maps with constants.

Contribution

It demonstrates that no bounded-length identities hold in a metric sense for symmetric groups and characterizes all self-maps of finite simple groups as word maps with constants.

Findings

No bounded-length identities hold in a metric sense for symmetric groups.

Word maps with constants of short length have images with positive diameter.

Every self-map of a finite non-abelian simple group is a word map with constants.

Abstract

We study word maps with constants on symmetric groups. Even though there are mixed identities of bounded length that are valid for all symmetric groups, we show that no such identities hold in a metric sense. Moreover, we prove that word maps with constants and non-trivial content that are short enough have an image of positive diameter only depending on the length of the word. Finally, we also show that every self-map $G \to G$ on a finite non-abelian simple group is actually a word map with constants from $G$.