# derived subgroup

Let $G$ be a group. For any $a,b\in G$, the element $a^{-1}b^{-1}ab$ is called the commutator of $a$ and $b$.

The commutator $a^{-1}b^{-1}ab$ is sometimes written $[a,b]$. (Usage varies, however, and some authors instead use $[a,b]$ to represent the commutator $aba^{-1}b^{-1}$.) If $A$ and $B$ are subsets of $G$, then $[A,B]$ denotes the subgroup of $G$ generated by $\{[a,b]\mid a\in A\hbox{ and }b\in B\}$. This notation can be further extended by recursively defining $[X_{1},\dots,X_{n+1}]=[[X_{1},\dots,X_{n}],X_{n+1}]$ for subsets $X_{1},\dots,X_{n+1}$ 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 $G^{\prime}$, or sometimes $G^{(1)}$.

Note that $a$ and $b$ commute if and only if the commutator of $a,b\in G$ is trivial, i.e.,

 $a^{-1}b^{-1}ab=1.$

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

###### 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 abelianization of $G$.

One can of course form the derived subgroup of the derived subgroup; this is called the second derived subgroup, and denoted by $G^{\prime\prime}$ or $G^{(2)}$. Proceeding inductively one defines the $n^{\text{th}}$ 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 $G$:

 $G=G^{(0)}\supseteq G^{(1)}\supseteq G^{(2)}\supseteq\cdots$
###### Proposition 2

The group $G$ is solvable if and only if the derived series terminates 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.

 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