Identity is only Idempotent Element in Group

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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.


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
\(\ds \) \(=\) \(\ds x \circ e\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds x\) Definition of Identity Element