Identity is only Idempotent Element in Group/Proof 2

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Theorem

Every group has exactly one idempotent element: the identity.


Proof

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

$\blacksquare$


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