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单词 DerivedSubgroup
释义

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.,

a-1b-1ab=1.

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 (http://planetmath.org/QuotientGroup) 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 derivedsubgroup G(n) as the derived subgroup of G(n-1). In this fashion oneobtains a sequence of subgroups, called the derived series of G:

G=G(0)G(1)G(2)
Proposition 2

The group G is solvable if and only if the derived seriesterminates in the trivial group {1} after a finite (http://planetmath.org/Finite) number of steps.

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

Titlederived subgroup
Canonical nameDerivedSubgroup
Date of creation2013-03-22 12:33:53
Last modified on2013-03-22 12:33:53
Owneryark (2760)
Last modified byyark (2760)
Numerical id22
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F14
Classificationmsc 20E15
Classificationmsc 20A05
Synonymcommutator subgroup
Related topicJordanHolderDecomposition
Related topicSolvable
Related topicTransfiniteDerivedSeries
Related topicAbelianization
Definescommutator
Definesderived series
Definessecond derived subgroup
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更新时间:2025/5/4 3:02:15