# Homomorphism commutes with word maps

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## Statement

Suppose $w$ is a word in the letters $x_1,x_2,\dots,x_n$ (these are just formal symbols). Suppose $\varphi:G \to H$ is a homomorphism of groups. Then, $\varphi$ commutes with $w$, i.e.:

$\varphi(w(g_1,g_2,\dots,g_n)) = w(\varphi(g_1),\varphi(g_2),\dots,\varphi(g_n)) \ \forall \ g_1,g_2,\dots,g_n \in G$

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

## Proof

A formal proof can be given by inducting on the length of the word. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]