Of course, G' is equal to {1} if and only if the group G is abelian (that is, for all g,g' in G, [g,g' ] = 1). In the general case this group, in a sense, gives a measure of how far G is from being abelian; the larger G' , the "less abelian" G is. As a subgroup of G, G' is normal, and the quotient G/G' is an abelian group called sometimes G made abelian; more in general, if a factor group G/N of G is abelian, it means that N includes G' .
A group is called perfect if it is equal to its derived group.