Easy# Computing the Order of an Element in a Group.

ABSALG-SVGA68

Let $G$ be a group with neutral element $e$. An element $g\in G$ is said to have finite order if we can compose $g$ a finite number of times with itself and get $e$.

When $g$ has finite order, the order of $g$ is defined to be the order of the cyclic subgroup $\langle g\rangle$ of $G$, generated by $g$. It is the *smallest positive* integer $k$ such that $g$ composed with itself $k$ times equals $e$.

If the cyclic subgroup $\langle g\rangle$ of $G$ has infinite order, then we say that the order of $g$ is infinite.

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