Derived group

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In mathematics, the derived group of a group G is the group G' generated by all the commutators of elements of G; that is, G' = <[g,g' ] : g,g' in G>.

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.