derived subgroup
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 definingfor 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.
Proposition 1
The derived subgroup is normal (in fact, fully invariant) in ,and the factor group is abelian.Moreover, is abelian if and only if is the trivial subgroup.
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 derivedsubgroup as the derived subgroup of . In this fashion oneobtains a sequence of subgroups, called the derived series of :
Proposition 2
The group is solvable if and only if the derived seriesterminates in the trivial group 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.
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 |