Identity of Group is Unique/Proof 3

Theorem

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

Then $e$ is unique.

Proof

From Group has Latin Square Property, there exists a unique $x \in G$ such that:

$a x = b$

and there exists a unique $y \in G$ such that:

$y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:

$a x = a$

and there exists a unique $y \in G$ such that:

$y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Proposition $3.3$: Remark $3$