I have done practice problems like this in $\mathbb{Z}_{12}$ to $\mathbb{Z}_{10}$ defined via $x \mapsto 3x$, but when it comes to $S_4$ I get tripped up. Can someone help me understand the process?
2026-03-27 19:53:28.1774641208
Define $\varphi:S_4\to S_4$ via $\alpha\mapsto(1 3 4)\alpha$. How would I prove $\varphi$ is not a homorphism?
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Hint: Every group homomorphism maps the identity element into the identity element.