Order of a power

Statement

Let $G$ be a group and let $g \in G$ be an element of finite order $n$. Let $k$ be an integer. Then the order of $g^k$ is $n / \gcd(n, k)$.

Remarks

Here $\gcd(n, 0) = n$, so that $g^0 = e$ has order $1$. In particular, $g^k$ has order $n$ if and only if $\gcd(n, k) = 1$. For example, if $g$ has order $12$, then $g^8$ has order $12 / 4 = 3$.

Explanations

L1 human

A power of an element returns to the identity faster than the element itself, or at the same time. If $g$ has order $12$, then $g^8$ is at the identity after $k$ steps exactly when $12$ divides $8k$, that is, when $3$ divides $k$. Thus $g^8$ has order $3$.