# Boolean Group is Abelian

## Theorem

Let $G$ be a Boolean group.

Then $G$ is abelian.

## Proof 1

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.

$\blacksquare$

## Proof 2

Let $a, b \in G$.

By definition of Boolean group:

$\forall x \in G: x^2 = e$

where $e$ is the identity of $G$.

Then:

 $\ds a b$ $=$ $\ds a e b$ Group Axiom $G2$: Properties of Identity $\ds$ $=$ $\ds a \left({a b}\right)^2 b$ as $\forall x \in G: x^2 = e$ $\ds$ $=$ $\ds a \left({a b}\right) \left({a b}\right) b$ $\ds$ $=$ $\ds \left({a a}\right) \left({b a}\right) \left({b b}\right)$ Group Axiom $G1$: Associativity $\ds$ $=$ $\ds a^2 \left({b a}\right) b^2$ $\ds$ $=$ $\ds e \left({b a}\right) e$ as $\forall x \in G: x^2 = e$ $\ds$ $=$ $\ds b a$

Thus $a b = b a$ and therefore $G$ is abelian.

$\blacksquare$