# 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 $\struct {G, \circ}$ be a group whose identity is $e$.

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

 $\ds e$ $=$ $\ds x \circ x^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds$ $=$ $\ds \paren {x \circ x} \circ x^{-1}$ by hypothesis: $x \circ x = x$ $\ds$ $=$ $\ds x \circ \paren {x \circ x^{-1} }$ Group Axiom $\text G 1$: Associativity $\ds$ $=$ $\ds x \circ e$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds$ $=$ $\ds x$ Group Axiom $\text G 2$: Existence of Identity Element

$\blacksquare$