Identity of Group is Unique/Proof 2

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Theorem

Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$.

Then $e$ is unique.


Proof

Let $e$ and $f$ both be identity elements of a group $\struct {G, \circ}$.

Then:

\(\displaystyle e\) \(=\) \(\displaystyle e \circ f\) $f$ is an identity
\(\displaystyle \) \(=\) \(\displaystyle f\) $e$ is an identity

So $e = f$ and there is only one identity after all.

$\blacksquare$


Sources