= Homomorphism from a finite cyclic group
{title2=$\operatorname{Hom}(C_n,G)\cong\{g\in G:g^n=1\}$}
= Homomorphisms from a finite cyclic group
{synonym}
For any <group> $G$, evaluation at a chosen <generator of a group> $x$ of $C_n$ gives a bijection between <group homomorphisms> $C_n\to G$ and elements satisfying $g^n=1$. A <group homomorphism> must send $x^j$ to $g^j$. Conversely, this formula is well-defined because exponents differing by a multiple of $n$ give equal powers, and it respects multiplication. The displayed correspondence is a bijection of sets; it is not claimed to be a group isomorphism for a nonabelian target. In a <symmetric group>, the <order of a group element> is the <least common multiple> of its disjoint cycle lengths, making this correspondence easy to enumerate.
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