# If a group has only one commutator, why does that mean it is abelian?

I understand that if $aba^{-1}b^{-1} = e$ then $ab$ is commutative, but I don't see how having multiple commutators will prevent the group from being abelian

• If $aba^{-1}b^{-1} = c \ne e$, then $ab=cba$, so $ab \ne ba$. – Derek Holt Feb 24 '14 at 16:10
• such a simple connection that I couldn't make, thank you!!! – Rod Feb 24 '14 at 16:14

It's easier to see the contrapositive: If a group is abelian, then $e$ is its only commutator:
$$aba^{-1}b^{-1} = aa^{-1}bb^{-1} = ee = e$$
Therefore, if the group has more than one commutator, at least one of them will be different from $e$, and so the group cannot be abelian.