Identity of Subgroup

Theorem
Every subgroup $H$ of a group $G$ contains the identity of $G$, which is also the identity of $H$.

Proof
From the definition, a group is a monoid.

Also, all of its elements are cancellable.

The result then follows directly from the result for monoids each of whose elements are cancellable: Identity of Cancellable Monoid is Identity of Submonoid.

Also see

 * Identity of Submonoid is not necessarily Identity of Monoid