Homomorphism commutes with word maps

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 $H$.

Related facts

Applications

• Verbal implies fully invariant

Proof

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

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