Question How to Count Homomorphism from any finite group G to infinite cyclic group
MY Approach
I Know
I also know that a homomorphism is completely determined by its action on unit element (like 1)
2026-02-22 21:31:55.1771795915
How to Count Homomorphism from any finite group G to infinite cyclic group?
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If $x\in G$ is not the identity, there is some number $m>1$ such that $x^m$ is the identity. Since a homomorphism always maps the identity to the identity, a homomorphism $f:G'\to G$ would have $f(x^m)=f(x)^m=e$. But this means that $f(x)=e$ because the only element of finite order in an infinite cyclic group is the identity.