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: Cancellable Monoid Identity of Submonoid.