Equivalent Characterizations of Abelian Group

Theorem

Let $G$ be a group.

The following statements are equivalent:

\((1)\)	$:$							$G$ is abelian
\((2)\)	$:$							$\forall a, b \in G: \paren {a b}^{-1} = a^{-1} b^{-1}$
\((3)\)	$:$							Definition of Cross Cancellation Property: $\forall a, b, c \in G: a b = c a \implies b = c$
\((4)\)	$:$							Definition of Middle Cancellation Property: $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$
\((5)\)	$:$							the opposite group to $G$ is $G$ itself.

Proof

$(1)$ iff $(2)$

See Inversion Mapping is Automorphism iff Group is Abelian.

$\Box$

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$(1)$ iff $(3)$

See Group is Abelian iff it has Cross Cancellation Property.

$\Box$

$(1)$ iff $(4)$

See Group is Abelian iff it has Middle Cancellation Property.

$\Box$

$(1)$ iff $(5)$

See Group is Abelian iff Opposite Group is Itself.

$\Box$

Hence all $5$ statements are logically equivalent.

$\blacksquare$