# Equivalent Characterizations of Abelian Group

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## Contents

## Theorem

Let $G$ be a group.

The following are equivalent:

\((1):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | $G$ is abelian | |||||||||

\((2):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | $\forall a, b \in G: \paren {a b}^{-1} = a^{-1} b^{-1}$ | |||||||||

\((3):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | Cross cancellation property: $\forall a, b, c \in G: a b = c a \implies b = c$ | |||||||||

\((4):\quad\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | 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

See Inversion Mapping is Automorphism iff Group is Abelian.

$\Box$

### 1 iff 3

See Group Abelian iff Cross Cancellation Property.

$\Box$

### 1 iff 4

See Group Abelian iff Middle Cancellation Property.

$\Box$

Hence all four statements are logically equivalent.

$\blacksquare$