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.

Related facts

Applications

Proof

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