= Integer power of a group element
{title2=$x^n\quad(n\in\mathbb Z)$}
= Integer powers of a group element
{synonym}
For an element $x$ of a <group> with <identity element> $e$, define $x^0=e$, $x^n$ as a product of $n$ copies of $x$ for positive $n$, and $x^{-n}=(x^{-1})^n$. <Associativity> and cancellation of <inverse elements> give $x^{m+n}=x^mx^n$ and $(x^m)^n=x^{mn}$ for all <integers> $m,n$. Consequently $n\mapsto x^n$ is a <group homomorphism> from the <additive group of integers> onto the <cyclic subgroup> generated by $x$.
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