Equivalent Characterizations of Abelian Group

From ProofWiki
Jump to navigation Jump to search


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: $\forall a, b, c, d, x \in G: a x b = c x d \implies a b = c d$


1 iff 2

See Inversion Mapping is Automorphism iff Group is Abelian.


1 iff 3

See Group Abelian iff Cross Cancellation Property.


1 iff 4

See Group Abelian iff Middle Cancellation Property.


Hence all four statements are logically equivalent.