derived subgroup

Let G be a group. For any a,bG, the element a-1b-1ab is called the commutator of a and b.

The commutator a-1b-1ab is sometimes written [a,b]. (Usage varies, however, and some authors instead use [a,b] to represent the commutator aba-1b-1.) If A and B are subsets of G, then [A,B] denotes the subgroupMathworldPlanetmathPlanetmath of G generated by {[a,b]aA and bB}. This notation can be further extended by recursively defining [X1,,Xn+1]=[[X1,,Xn],Xn+1] for subsets X1,,Xn+1 of G.

The subgroup of G generated by all the commutators in G (that is, the smallest subgroup of G containing all the commutators) is called the derived subgroup, or the commutator subgroup, of G. Using the notation of the previous paragraph, the derived subgroup is denoted by [G,G]. Alternatively, it is often denoted by G, or sometimes G(1).

Note that a and b commute if and only if the commutator of a,bG is trivial, i.e.,


Thus, in a fashion, the derived subgroup measures the degree to which a group fails to be abelianMathworldPlanetmath.

Proposition 1

The derived subgroup [G,G] is normal (in fact, fully invariant) in G, and the factor group G/[G,G] is abelian. Moreover, G is abelian if and only if [G,G] is the trivial subgroup.

The factor group G/[G,G] is the largest abelian quotient ( of G, and is called the abelianizationMathworldPlanetmath of G.

One can of course form the derived subgroup of the derived subgroup; this is called the second derived subgroup, and denoted by G′′ or G(2). Proceeding inductively one defines the nth derived subgroup G(n) as the derived subgroup of G(n-1). In this fashion one obtains a sequence of subgroups, called the derived series of G:

Proposition 2

The group G is solvable if and only if the derived series terminates in the trivial group {1} after a finite ( number of steps.

The derived series can also be continued transfinitely—see the article on the transfinite derived series.

Title derived subgroup
Canonical name DerivedSubgroup
Date of creation 2013-03-22 12:33:53
Last modified on 2013-03-22 12:33:53
Owner yark (2760)
Last modified by yark (2760)
Numerical id 22
Author yark (2760)
Entry type Definition
Classification msc 20F14
Classification msc 20E15
Classification msc 20A05
Synonym commutator subgroup
Related topic JordanHolderDecomposition
Related topic Solvable
Related topic TransfiniteDerivedSeries
Related topic Abelianization
Defines commutator
Defines derived series
Defines second derived subgroup