Commutator subgroup

Definition

Let $G$ be a group. The commutator subgroup of $G$ is the subgroup generated by the set of all commutators $[g, h]$ with $g, h \in G$. It is written $[G, G]$ or $G'$.

Remarks

A product of commutators need not be a commutator, so in general the set of commutators is not a subgroup. The subgroup $[G, G]$ is characteristic, so it is normal in $G$. The group $G$ is abelian if and only if $[G, G] = \{e\}$. The commutator subgroup of the symmetric group $S_n$ is the alternating group $A_n$.

Explanations

L1 human

The commutator subgroup collects all the failures of commutativity in a group. If you divide the group by it, you get an abelian group. A large commutator subgroup means a group that is far from abelian.