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Let  be a group. For any  , the element
 is called the commutator of  and  .
The commutator
 is sometimes written ![$ [a,b]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img7.png "$ [a,b]$") . (Usage varies, however, and some authors instead use ![$ [a,b]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img8.png "$ [a,b]$") to represent the commutator
 .) If  and  are subsets of  , then ![$ [A,B]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img13.png "$ [A,B]$") denotes the subgroup of  generated by
![$ \{[a,b]\mid a\in A\hbox{ and }b\in B\}$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img15.png "$ \{[a,b]\mid a\in A\hbox{ and }b\in B\}$") . This notation can be further extended by recursively defining
![$ [X_1,\dots,X_{n+1}]=[[X_1,\dots,X_n],X_{n+1}]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img16.png "$ [X_1,\dots,X_{n+1}]=[[X_1,\dots,X_n],X_{n+1}]$") for subsets
 of  .
The subgroup of  generated by all the commutators in  (that is, the smallest subgroup of  containing all the commutators) is called the derived subgroup, or the commutator subgroup, of  . Using the notation of the previous paragraph, the derived subgroup is denoted by ![$ [G,G]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img23.png "$ [G,G]$") . Alternatively, it is often denoted by  , or sometimes  .
Note that  and  commute if and only if the commutator of  is trivial, i.e.,
Thus, in a fashion, the derived subgroup measures the degree to which a group fails to be abelian.
The factor group ![$ G/[G,G]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img35.png "$ G/[G,G]$") is the largest abelian quotient of  , and is called the abelianization of  .
One can of course form the derived subgroup of the derived subgroup; this is called the second derived subgroup, and denoted by  or  . Proceeding inductively one defines the
 derived subgroup  as the derived subgroup of  . In this fashion one obtains a sequence of subgroups, called the derived series of  :
Proposition 2 The group  is solvable if and only if the derived series terminates in the trivial group  after a finite number of steps.
The derived series can also be continued transfinitely--see the article on the transfinite derived series.
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![$ [G,G]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img30.png "$ [G,G]$"){kind=small}
![$ G/[G,G]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img32.png "$ G/[G,G]$"){kind=small}
![$ [G,G]$](http://images.planetmath.org:8080/cache/objects/2812/l2h/img34.png "$ [G,G]$"){kind=small}
