Identity of Subgroup

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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.

$\blacksquare$


Also see


Sources