Abelianisation

Definition

Let $G$ be a group and let $[G, G]$ be its commutator subgroup. The abelianisation of $G$ is the quotient group $G^{\mathrm{ab}} = G/[G, G]$.

Remarks

The group $G^{\mathrm{ab}}$ is abelian, by this criterion. If $G$ is abelian, then $G^{\mathrm{ab}} = G$. The abelianisation of the symmetric group $S_n$ for $n \geq 2$ is cyclic of order $2$, and the sign of a permutation gives the isomorphism.

Explanations

L1 human

The abelianisation forces all elements of a group to commute, and changes nothing more than it must. It keeps as much information about the group as an abelian group can keep. For a group that is already abelian, it does nothing.