Identity of Subgroup

Theorem
Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of group $G$.

Then the identity of $H$ is also $e$.

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