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 $ab{a}^{-1}{b}^{-1}$.) If $A$ and $B$ are subsets of $G$, then $[A,B]$ denotes the subgroup of $G$ generated by $\{[a,b]\mid a\in A\text{ and }b\in B\}$. This notation can be further extended by recursively defining $[X_1,\mathrm{\dots },X_{n+1}]=[[X_1,\mathrm{\dots },X_n],X_{n+1}]$ for subsets $X_1,\mathrm{\dots },X_{n+1}$ 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 $\mathrm{[}G\mathrm{,}G\mathrm{]}$ is normal (in fact, fully invariant) in $G$, and the factor group $G\mathrm{/}\mathrm{[}G\mathrm{,}G\mathrm{]}$ is abelian. Moreover, $G$ is abelian if and only if $\mathrm{[}G\mathrm{,}G\mathrm{]}$ 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 $G$:

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

Proposition 2

The group $G$ is solvable if and only if the derived series terminates in the trivial group $\mathrm{\{}\mathrm{1}\mathrm{\}}$ 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