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Homederived subgroup

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# 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 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 number of steps.

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

## Mathematics Subject Classification

20F14*no label found*20E15

*no label found*20A05

*no label found*

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