# 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 $\text G 2$: Existence of Identity Element $\ds$ $=$ $\ds a \paren {a b}^2 b$ as $\forall x \in G: x^2 = e$ $\ds$ $=$ $\ds a \paren {a b} \paren {a b} b$ $\ds$ $=$ $\ds \paren {a a} \paren {b a} \paren {b b}$ Group Axiom $\text G 1$: Associativity $\ds$ $=$ $\ds a^2 \paren {b a} b^2$ $\ds$ $=$ $\ds e \paren {b a} e$ as $\forall x \in G: x^2 = e$ $\ds$ $=$ $\ds b a$

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

$\blacksquare$