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A group that every element has order 2 is abelian


Let $G$ be a group. Prove that if $x^2 = 1$ for all $x \in G$, then $G$ is abelian.


Solution: Note that since $x^2 = 1$ for all $x \in G$, we have $x^{-1} = x$. Now let $a,b \in G$. We have $$ab = (ab)^{-1} = b^{-1} a^{-1} = ba.$$ Thus $G$ is abelian.

Linearity

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