Algebra Solutions: In a group, if every element has order 2 the group is abelian

In a group, if every element has order 2 the group is abelian

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

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.

Algebra Solutions

In a group, if every element has order 2 the group is abelian

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