Identity of Subsemigroup of Group

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Let $\struct {G, \circ}$ be a group whose identity element is $e$.

Let $\struct {H, \circ}$ be a subsemigroup of $\struct {G, \circ}$.

If $\struct {H, \circ}$ has an identity element, then that identity is $e$.


From the Cancellation Laws, every element of $\struct {G, \circ}$ is cancellable.

From Identity is only Idempotent Cancellable Element, there is only one element $x$ of $\struct {G, \circ}$ satisfying $x \circ x = x$, and that is $e$.

As $x \in H \implies x \in G$, the same applies to $\struct {H, \circ}$.

So if there is an element in $\struct {H, \circ}$ such that $x \circ x = x$, it must be $e$.

Hence the result.