Homomorphism commutes with word maps: Difference between revisions

Latest revision as of 16:50, 22 June 2012

Statement

Suppose $w$ is a word in the letters $x_{1}, x_{2}, \dots, x_{n}$ (these are just formal symbols). Suppose $φ : G \to H$ is a homomorphism of groups. Then, $φ$ commutes with $w$ , i.e.:

$φ (w (g_{1}, g_{2}, \dots, g_{n})) = w (φ (g_{1}), φ (g_{2}), \dots, φ (g_{n})) \forall g_{1}, g_{2}, \dots, g_{n} \in G$

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]

@@ Line 7: / Line 7: @@
 where the <math>w</math> on the left is the [[word map]] in <math>G</math> (i.e., it evaluates the word for a tuple of values of the letters in <math>G</math> and the <math>w</math> on the right is the word map in <math>H</math>.
+==Related facts==
+===Applications===
+* [[Verbal implies fully invariant]]
 ==Proof==
 A formal proof can be given by inducting on the length of the word. {{fillin}}