Boolean Group is Abelian

Theorem
Let $(G, \circ)$ be a group whose identity is $e$.

If $G$ is Boolean then it is abelian.

Proof
Group $G$ is Boolean which means that all its elements, besides identity, have order $2$.

Because every element of order $2$ is self-inverse and identity is also self-inverse by the virtue of $e \circ e = e$ then all elements of $G$ are self-inverse.

Now result holds due to All Self-Inverse then Abelian.