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Definition ($|a|$ and $|G|$ - Order For Elements and Groups)

Let $G$ be a group with identity $e$, and let $g\in G$.

  • The $\textdef{order}$ of $G$, denoted $|G|$, is the cardinality of $G$.
  • The $\textdef{order}$ of $g$, denoted $|g|$, is the smallest positive integer $n$ such that $g^n = e$, if such an $n$ exists. If no such $n$ exists, we say $g$ has infinite order.

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