Boolean Group is Abelian/Proof 1

Theorem

Let $G$ be a Boolean group.

Then $G$ is abelian.

Proof

By definition of Boolean group, all elements of $G$, other than the identity, have order $2$.

By Group Element is Self-Inverse iff Order 2 and Identity is Self-Inverse, all elements of $G$ are self-inverse.

The result follows directly from All Elements Self-Inverse then Abelian.

$\blacksquare$