All Elements Self-Inverse then Abelian

Theorem
Let $\left({G, \circ}\right)$ be a group.

Suppose that every element of $G$ is self-inverse.

Then $G$ is abelian.

Proof
Every element of $G$ is self-inverse, i.e.:


 * $\forall x \in G: x \circ x = e$

In particular, for all $x, y \in G$:


 * $\left({x \circ y}\right) \circ \left({x \circ y}\right) = e$

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, for all $x, y \in G$.

Hence $G$ is an abelian group.