Group element

A group element is a member of the underlying set of a group. The group operation assigns a product to each pair of elements, the identity fixes every element under multiplication, and every element has an inverse.

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 $x^{m+n}=x^m{x}^n$ 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 $x$.