# Identity of Group is Unique

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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 1

By the definition of a group, $\struct {G, \circ}$ is also a monoid.

The result follows by applying the result Identity of Monoid is Unique.

$\blacksquare$

## Proof 2

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

Then:

\(\ds e\) | \(=\) | \(\ds e \circ f\) | $f$ is an identity | |||||||||||

\(\ds \) | \(=\) | \(\ds f\) | $e$ is an identity |

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

$\blacksquare$

## Proof 3

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

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.6$. Elementary theorems on groups - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$