# 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 |