Identity of Subsemigroup of Group

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Definition

Let $\left({G, \circ}\right)$ be a group whose identity element is $e$.

Let $\left({H, \circ}\right)$ be a subsemigroup of $\left({G, \circ}\right)$.


If $\left({H, \circ}\right)$ has an identity element, then that identity is $e$.


Proof

From the Cancellation Laws, every element of $\left({G, \circ}\right)$ is cancellable.

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

As $x \in H \implies x \in G$, the same applies to $\left({H, \circ}\right)$.

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

Hence the result.

$\blacksquare$


Sources