If $G$ is an abelian group, then inverse of $x$ is equal to $x$?

Question

True/False

If $G$ is an abelian group, then the inverse of $x$ is equal to $x$, true or false and why?

Answer 1

Obviously false. Consider $\mathbb{Z}/3\mathbb{Z}$ with addition. Then

$$1+2 = 0 = 2+1$$

so $2$ is the inverse of $1$, yet $1 \neq 2$.

And there are plenty more counterexamples.

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Note: the converse to your statement does hold.

That is, if every element in a group is equal to its inverse, then the group is abelian. Indeed, if $G$ is such a group and $g,h \in G$, then

$$gh = g^{-1}h^{-1} = (hg)^{-1} = hg$$

An example of a group where every element is equal to its inverse is $\mathbb{Z}_2 \times \mathbb{Z}_2$, as @amWhy notices in the comments.