Identity Element is Unique

Theorem

In a Group $G$ the identity element is unique.

Proof

Suppose $e_{1}, e_{2} \in G$ are both identity elements. Then:

e_{1} ⋆ e_{2} = e_{2}

because $e_{1}$ is an identity element. Similarly $e_{1} ⋆ e_{2} = e_{2}$ since $e_{2}$ is an identity. Thus $e_{1} = e_{2}$ .

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