# Definition The order of a [[group]] $G$ is the number of [[Group Element|elements]] in a group. It is denoted by $ord(G)$ or $|G|$. # Theorem The order of a [[subgroup]] is a divisior of the order of the group. ### Proof Let the group be $G$ and the subgroup be $H$. Let $g \in H$. We define a [[Coset]] $gH$. Let the number of distinct cosets be $n$. $n$ multiplied by the number of elements of $gH$ is $|G|$. This is because, since *distinct* cosets have no elements in common, the $n$ distinct cosets we have contain all the elements, and multiplying $n$ by the number of elements in any of them gives $|G|$.