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

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

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

$\blacksquare$


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