Editing Commutator subgroup (section)

{{short description|Smallest normal subgroup by which the quotient is commutative}}
In [[mathematics]], more specifically in [[abstract algebra]], the '''commutator subgroup''' or '''derived subgroup''' of a [[group (mathematics)|group]] is the [[subgroup (mathematics)|subgroup]] [[generating set of a group|generated]] by all the [[commutator]]s of the group.<ref>{{harvtxt|Dummit|Foote|2004}}</ref><ref>{{harvtxt|Lang|2002}}</ref>

The commutator subgroup is important because it is the [[Universal property|smallest]] [[normal subgroup]] such that the [[quotient group]] of the original group by this subgroup is [[abelian group|abelian]]. In other words, <math>G/N</math> is abelian [[if and only if]] <math>N</math> contains the commutator subgroup of <math>G</math>. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.