Derived series - Revision history

Vipul: /* Subgroup properties satisfied by members */

2026-05-03T07:23:00Z

Subgroup properties satisfied by members

← Older revision		Revision as of 07:23, 3 May 2026
Line 36:		Line 36:
	===Subgroup properties satisfied by members===		===Subgroup properties satisfied by members===

	The [[derived series]] is a [[strongly fully invariant series]] and hence a [[strongly characteristic series]]: every smaller member of the series is a [[fully invariant subgroup]] in every bigger (earlier) member, and hence also a [[characteristic subgroup]] in every bigger (earlier) member. Each member is a [[fully ~~characteristic~~ subgroup]]; in fact, each member is an [[image-closed fully invariant subgroup]]. The finite members are also [[verbal subgroup]]s.		The [[derived series]] is a [[strongly fully invariant series]] and hence a [[strongly characteristic series]]: every smaller member of the series is a [[fully invariant subgroup]] in every bigger (earlier) member, and hence also a [[characteristic subgroup]] in every bigger (earlier) member. Each member is a [[fully invariant subgroup]]; in fact, each member is an [[image-closed fully invariant subgroup]]. The finite members are also [[verbal subgroup]]s.

	===Related group properties===		===Related group properties===

Vipul: /* For a solvable group */

2026-05-03T07:22:36Z

For a solvable group

← Older revision		Revision as of 07:22, 3 May 2026
Line 28:		Line 28:
	===For a solvable group===		===For a solvable group===

	A group is solvable if and only if its derived series terminates in finitely many steps at the identity element. The smallest <math>l</math> for which <math>G^{(l)}</math> is the trivial group is termed the ~~solvable~~ length ~~of the derived series~~. The derived series is the fastest descending series with the property that all successive quotients are Abelian. Note that there is ''no'' single series that serves the role of the fastest ''ascending'' series with all successive quotients Abelian.		A group is solvable if and only if its derived series terminates in finitely many steps at the identity element. The smallest <math>l</math> for which <math>G^{(l)}</math> is the trivial group is termed the [[derived length]]. The derived series is the fastest descending series with the property that all successive quotients are Abelian. Note that there is ''no'' single series that serves the role of the fastest ''ascending'' series with all successive quotients Abelian.

	{{further\|[[Derived series is fastest descending series with Abelian quotients]]}}		{{further\|[[Derived series is fastest descending series with Abelian quotients]]}}

Vipul: /* Subgroup properties satisfied by members */

2013-02-17T04:11:43Z

Subgroup properties satisfied by members

← Older revision		Revision as of 04:11, 17 February 2013
Line 36:		Line 36:
	===Subgroup properties satisfied by members===		===Subgroup properties satisfied by members===

	The [[derived series]] is a [[strongly fully invariant series]] and hence a [[strongly characteristic series]]: every smaller member of the series is a fully characteristic subgroup in every bigger member. Each member is a [[fully characteristic subgroup]]; in fact, each member is an [[image-closed fully ~~characteristic~~ subgroup]]. The finite members are also [[verbal subgroup]]s.		The [[derived series]] is a [[strongly fully invariant series]] and hence a [[strongly characteristic series]]: every smaller member of the series is a [[fully invariant subgroup]] in every bigger (earlier) member, and hence also a [[characteristic subgroup]] in every bigger (earlier) member. Each member is a [[fully characteristic subgroup]]; in fact, each member is an [[image-closed fully invariant subgroup]]. The finite members are also [[verbal subgroup]]s.

	===Related group properties===		===Related group properties===

Vipul: /* Facts */

2013-02-17T04:11:01Z

Facts

← Older revision		Revision as of 04:11, 17 February 2013
Line 36:		Line 36:
	===Subgroup properties satisfied by members===		===Subgroup properties satisfied by members===

	The [[derived series]] is a [[strongly fully characteristic series]]: every smaller member of the series is a fully characteristic subgroup in every bigger member. Each member is a [[fully characteristic subgroup]]; in fact, each member is an [[image-closed fully characteristic subgroup]]. The finite members are also [[verbal subgroup]]s.		The [[derived series]] is a [[strongly fully invariant series]] and hence a [[strongly characteristic series]]: every smaller member of the series is a fully characteristic subgroup in every bigger member. Each member is a [[fully characteristic subgroup]]; in fact, each member is an [[image-closed fully characteristic subgroup]]. The finite members are also [[verbal subgroup]]s.

	===Related group properties===		===Related group properties===

Vipul: /* Definition with symbols */

2013-02-17T04:10:33Z

Definition with symbols

← Older revision		Revision as of 04:10, 17 February 2013
Line 18:		Line 18:
	* <math>G^{(0)} = G</math>.		* <math>G^{(0)} = G</math>.
	* <math>G^{(\alpha + 1)} = [G^\alpha,G^\alpha]</math>.		* <math>G^{(\alpha + 1)} = [G^\alpha,G^\alpha]</math>.
	* <math>G^\gamma = \bigcap_{\alpha < \gamma} G_\alpha</math> for any limit ordinal <math>\gamma</math>.		* <math>G^{(\gamma)} = \bigcap_{\alpha < \gamma} G_\alpha</math> for any limit ordinal <math>\gamma</math>.

	Often the term is used to refer ''only'' to the finite part of the series:		Often the term is used to refer ''only'' to the finite part of the series:

Vipul: New page: {{iterated series|commutator subgroup}} ==Definition== ===Symbol-free definition=== The '''derived series''' or '''commutator series''' of a group is defined as follows: * The zero...

2008-10-09T22:46:27Z

New page: {{iterated series|commutator subgroup}} ==Definition== ===Symbol-free definition=== The '''derived series''' or '''commutator series''' of a group is defined as follows: * The zero...

New page

{{iterated series|commutator subgroup}}

==Definition==

===Symbol-free definition===

The '''derived series''' or '''commutator series''' of a [[group]] is defined as follows:

* The zeroth member is the group itself.
* The member indexed by a successor ordinal is the [[defining ingredient::commutator subgroup]] of its predecessor.
* The member indexed by a limit ordinal is the intersection of all previous members.

The intersection of all members of the derived series of a group equals its [[perfect core]].
===Definition with symbols===

The derived series of a group <math>G</math> indexed <math>G^{(\alpha)}</math>, is defined as follows:

* <math>G^{(0)} = G</math>.
* <math>G^{(\alpha + 1)} = [G^\alpha,G^\alpha]</math>.
* <math>G^\gamma = \bigcap_{\alpha < \gamma} G_\alpha</math> for any limit ordinal <math>\gamma</math>.

Often the term is used to refer ''only'' to the finite part of the series:

<math>G^{(0)} = G \ge G^{(1)} = [G,G] \ge G^{(2)} = [[G,G],[G,G]] \ge \dots</math>.

The intersection of ''all'' members of the derived series of a group equals its [[perfect core]].

===For a solvable group===

A group is solvable if and only if its derived series terminates in finitely many steps at the identity element. The smallest <math>l</math> for which <math>G^{(l)}</math> is the trivial group is termed the solvable length of the derived series. The derived series is the fastest descending series with the property that all successive quotients are Abelian. Note that there is ''no'' single series that serves the role of the fastest ''ascending'' series with all successive quotients Abelian.

{{further|[[Derived series is fastest descending series with Abelian quotients]]}}

==Facts==

===Subgroup properties satisfied by members===

The [[derived series]] is a [[strongly fully characteristic series]]: every smaller member of the series is a fully characteristic subgroup in every bigger member. Each member is a [[fully characteristic subgroup]]; in fact, each member is an [[image-closed fully characteristic subgroup]]. The finite members are also [[verbal subgroup]]s.

===Related group properties===

* [[Perfect group]]: A group whose derived series terminates at the group itself: it equals its commutator subgroup.
* [[Solvable group]]: A group whose derived series terminates in finitely many steps at the trivial subgroup.
* [[Hypoabelian group]]: A group whose transfinite derived series terminates at the trivial subgroup. Equivalently, it is a group whose perfect core is trivial.
* [[Residually solvable group]]: A group such that the intersection of all finite members of its derived series is the trivial subgroup. Equivalently, for every element of the group, there is a normal subgroup of finite index not containing that element such that the quotient by that subgroup is a solvable group.