All Elements Self-Inverse then Abelian

Theorem
If every element in a group is self-inverse, then that group is abelian.

Proof
If every element in a group $$\left({G, \circ}\right)$$ is self-inverse, then $$\forall x \in G: x \circ x = e$$.

Also, for all $$\forall x, y \in G: \left({x \circ y}\right) \circ \left({x \circ y}\right) = e$$, that is, $$x \circ y$$ is also self-inverse.

From Self-Inverse Elements that Commute, it follows that $$y \circ x = x \circ y$$, i.e. that $$x$$ and $$y$$ commute.

As this is true for all elements of $$\left({G, \circ}\right)$$, it follows that all elements of $$\left({G, \circ}\right)$$ commute with all other elements of $$\left({G, \circ}\right)$$.

Thus, by definition, $$\left({G, \circ}\right)$$ is abelian.