Equivalent Characterizations of Abelian Group

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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.

$\blacksquare$