# Identity is only Idempotent Element in Group

## Theorem

Every group has exactly one idempotent element: the identity.

## Proof 1

From the Cancellation Laws, all group elements are cancellable.

The result follows from Identity is only Idempotent Cancellable Element.

$\blacksquare$

## Proof 2

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $x \in G$ such that $x \circ x = x$.

 $\displaystyle e$ $=$ $\displaystyle x \circ x^{-1}$ Group axiom $G3$: every element has an inverse $\displaystyle$ $=$ $\displaystyle \left({x \circ x}\right) \circ x^{-1}$ by hypothesis: $x \circ x = x$ $\displaystyle$ $=$ $\displaystyle x \circ \left({x \circ x^{-1} }\right)$ Group axiom $G1$: $\circ$ is associative $\displaystyle$ $=$ $\displaystyle x \circ e$ Group axiom $G3$: $x^{-1}$ is the inverse of $x$ $\displaystyle$ $=$ $\displaystyle x$ Group axiom $G2$: $e$ is the identity of $G$

$\blacksquare$