# Homomorphism commutes with word maps

## Contents

## Statement

Suppose is a word in the letters (these are just formal symbols). Suppose is a homomorphism of groups. Then, commutes with , i.e.:

where the on the left is the word map in (i.e., it evaluates the word for a tuple of values of the letters in and the on the right is the word map in .

## Related facts

### Applications

## Proof

A formal proof can be given by inducting on the length of the word.

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