= Cyclic group
{title2=$C_n$}
{wiki}
A <cyclic group> is a <group> that can be generated by repeating a single <group element> $g$ and its <inverse element>:
$$
G=\langle g\rangle=\{g^k:k\in\mathbb Z\}.
$$
For an additive <group>, the same expression is written $\{kg:k\in\mathbb Z\}$: $kg$ means repeated <addition>, including subtraction when $k$ is negative. A <finite cyclic group> has a repeating cycle, while an <infinite cyclic group> is isomorphic to the additive <integers>. The notation $C_n$ refers to a <finite cyclic group> of order $n$.
This definition compresses a whole composition rule into one repeatable operation. Once $g$ is chosen, $g^ag^b=g^{a+b}$, so working with <group elements> becomes working with <integer> exponents. In the finite case those exponents are reduced using <modular arithmetic>. It also proves that every <cyclic group> is an <abelian group>, because $a+b=b+a$. A <finite cyclic group> of order $n$ has exactly $d$ solutions of $x^d=e$ for every positive <integer divisor> $d$ of $n$: in exponents, $n\mid dk$ has exactly $d$ solutions modulo $n$. The standard finite and infinite models are described in <Group Theory (v4.01) by James S. Milne> (section 1.16).
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