Let be a group. For any , the element is called the commutator of and .
The commutator is sometimes written . (Usage varies, however, and some authors instead use to represent the commutator .) If and are subsets of , then denotes the subgroup of generated by . This notation can be further extended by recursively defining for subsets of .
The subgroup of generated by all the commutators in (that is, the smallest subgroup of containing all the commutators) is called the derived subgroup, or the commutator subgroup, of . Using the notation of the previous paragraph, the derived subgroup is denoted by . Alternatively, it is often denoted by , or sometimes .
Note that and commute if and only if the commutator of is trivial, i.e.,
Thus, in a fashion, the derived subgroup measures the degree to which a group fails to be abelian.
The factor group is the largest abelian quotient (http://planetmath.org/QuotientGroup) of , and is called the abelianization of .
One can of course form the derived subgroup of the derived subgroup; this is called the second derived subgroup, and denoted by or . Proceeding inductively one defines the derived subgroup as the derived subgroup of . In this fashion one obtains a sequence of subgroups, called the derived series of :
The derived series can also be continued transfinitely—see the article on the transfinite derived series.
|Date of creation||2013-03-22 12:33:53|
|Last modified on||2013-03-22 12:33:53|
|Last modified by||yark (2760)|
|Defines||second derived subgroup|