Let $G$ be a finite group and $g \in G$. Then the order of $g$ is finite.

Question

Let $G$ be a finite group and $g \in G$. Then the order of $g$ is finite.

I know that if $g$ has finite order, then $g^k = e$ for a finite $k$, but i'm not really sure how to get started showing this. Since there is a finite number of elements in $G$, it makes intuitive sense that this result should be true.

Answer 1

Hint: consider the infinite sequence of elements $g,g^2,g^3\dots$. It must contain repeated elements $g^l=g^k$ for $l<k$. So we have $eg^l=g^{k-l}g^l$, what can we conclude about $g^{k-l}$ if we use the cancellation law?