Identity is only Idempotent Element in Group

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Theorem

Every group has exactly one idempotent element: the identity.


Proof 1

The Identity Element is Idempotent.

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 $G$

$\blacksquare$