g to the order of the group is the identity

Statement

Let $G$ be a finite group with identity element $e$, and let $g \in G$. Then $g^{|G|} = e$.

Remarks

The order of $g$ divides $|G|$ by this corollary, and a power is $e$ when the order divides the exponent. For the group $(\mathbb{Z}/p\mathbb{Z})^\times$ of order $p - 1$, with $p$ prime, this gives Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ for $p \nmid a$. For the group $(\mathbb{Z}/n\mathbb{Z})^\times$ of order $\varphi(n)$, it gives Euler's theorem.

Explanations

L1 human

If a group has $n$ elements, then the $n$th power of each element is the identity. For example, in a group of order $6$, each element $g$ satisfies $g^6 = e$. This simple fact proves Fermat's little theorem in number theory.