# Equivalent Characterizations of Abelian Group

## Theorem

Let $G$ be a group.

The following are equivalent:

 $\text {(1)}: \quad$ $\displaystyle$  $\displaystyle$ $G$ is abelian $\text {(2)}: \quad$ $\displaystyle$  $\displaystyle$ $\forall a, b \in G: \paren {a b}^{-1} = a^{-1} b^{-1}$ $\text {(3)}: \quad$ $\displaystyle$  $\displaystyle$ Cross cancellation property: $\forall a, b, c \in G: a b = c a \implies b = c$ $\text {(4)}: \quad$ $\displaystyle$  $\displaystyle$ Middle cancellation property: $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$

## Proof

### 1 iff 2

$\Box$

### 1 iff 3

$\Box$

### 1 iff 4

$\Box$

Hence all four statements are logically equivalent.

$\blacksquare$