Identity of Subgroup

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

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

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


A group is a fortiori a monoid.

From the Cancellation Laws, all of its elements are cancellable.

The result then follows from Identity of Cancellable Monoid is Identity of Submonoid.


Also see