# Identity is Only Group Element of Order 1

## Theorem

In every group, the identity, and only the identity, has order $1$.

## Proof

Let $G$ be a group with identity $e$.

Then:

$e^1 = e$

and:

$\forall a \in G: a \ne e: a^1 = a \ne e$.

Hence the result.

$\blacksquare$