Solvable group

Definition

Solvable is also called soluble by some people.

Equivalent definitions in tabular format

No.	Shorthand	A group is termed solvable if ...	A group $G$ is termed solvable if ...
1	normal series, abelian quotients	there is a normal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group.	there exists a series of subgroups $\{e\}=H_{0}\leq H_{1}\leq \ldots \leq H_{n}=G$ such that each $H_{i}$ is normal in $G$ and each $H_{i+1}/H_{i}$ is abelian.
2	subnormal series, abelian quotients	there is a subnormal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group.	there exists a series of subgroups:$\{e\}=H_{0}{\underline {\triangleleft }}H_{1}{\underline {\triangleleft }}\dots {\underline {\triangleleft }}H_{n}=G$ such that each $H_{i}$ is normal in $H_{i+1}$ and each $H_{i+1}/H_{i}$ is abelian.
3	derived series finite length	the derived series reaches the identity in finitely many steps	the derived series of $G$, i.e., the series $G^{(n)}$ where $G^{(0)}=G$ and $G^{(i+1)}=[G^{(i)},G^{(i)}]$ is the derived subgroup of its predecessor, reaches the trivial subgroup in finitely many steps.
4	characteristic series, abelian quotients	there is a characteristic series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group.	there exists a series of subgroups $\{e\}=H_{0}\leq H_{1}\leq \ldots \leq H_{n}=G$ such that each $H_{i}$ is characteristic in $G$ and each $H_{i+1}/H_{i}$ is abelian.
5	fully invariant series, abelian quotients	there is a fully invariant series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group.	there exists a series of subgroups $\{e\}=H_{0}\leq H_{1}\leq \ldots \leq H_{n}=G$ such that each $H_{i}$ is fully invariant in $G$ and each $H_{i+1}/H_{i}$ is abelian.

The length of the derived series, and the smallest possible length of a series for any of the other equivalent definitions, is termed the derived length or solvable length of the group.

Equivalence of definitions

Further information: Equivalence of definitions of solvable group, equivalence of definitions of derived length

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Particular note-worthy examples are given below:

• The symmetric group of degree three is the smallest solvable non-abelian(in fact, non-nilpotent) group
• The symmetric group of degree four is also solvable.
• The dihedral group of any order is solvable, Further it is nilpotent only when the order is a power of 2.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Solvable group, all facts related to Solvable group) |Survey articles about this | Survey articles about definitions built on this
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View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

The version of this for finite groups is at: finite solvable group

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
extension-closed group property	Yes	solvability is extension-closed	Suppose $H$ is a normal subgroup of $G$ such that both $H$ and the quotient group $G/H$ are solvable groups. Then $G$ is a solvable group.
quasivarietal group property	Yes	solvability is quasivarietal	subgroup-closed, quotient-closed, and closed under finite direct products (see items below)
subgroup-closed group property	Yes	solvability is subgroup-closed	If $G$ is solvable, and $H\leq G$ is a subgroup, then $H$ is solvable.
quotient-closed group property	Yes	solvability is quotient-closed	If $G$ is solvable, and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is solvable.
finite direct product-closed group property	Yes	solvability is finite direct product-closed	If $G_{1},G_{2},\times ,G_{n}$ are solvable, the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$ is also solvable.
finite normal join-closed group property	Yes	solvability is finite normal join-closed	If $G$ is a group and $N_{1},N_{2},\dots ,N_{r}$ are all solvable normal subgroups of $G$, the join of subgroups (in this case also the product of subgroups) $N_{1}N_{2}\dots N_{r}$ is also solvable.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions	Comparison
abelian group	commutator subgroup is trivial	abelian implies solvable	solvable not implies abelian (see also list of examples)
cyclic group			(see also list of examples)
nilpotent group	lower central series reaches the identity	nilpotent implies solvable	solvable not implies nilpotent (see also list of examples)	|FULL LIST, MORE INFO	nilpotent versus solvable
metabelian group	abelian normal subgroup with abelian quotient; derived length two		(see also list of examples)	|FULL LIST, MORE INFO
supersolvable group	normal series with cyclic factor groups	supersolvable implies solvable	solvable not implies supersolvable (see also list of examples)	|FULL LIST, MORE INFO
polycyclic group	subnormal series with cyclic factor groups	polycyclic implies solvable	solvable not implies polycyclic (see also list of examples)	|FULL LIST, MORE INFO
metacyclic group	cyclic normal subgroup with cyclic quotient group		(see also list of examples)	For intermediate notions between solvable group and metacyclic group, click here.

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions	Comparison
hypoabelian group	transfinite derived series reaches identity	solvable implies hypoabelian	hypoabelian not implies solvable	|FULL LIST, MORE INFO
imperfect group	no nontrivial perfect quotient group	solvable implies imperfect	imperfect not implies solvable	|FULL LIST, MORE INFO
locally solvable group	every finitely generated subgroup is solvable
residually solvable group	every non-identity element has a non-identity image in some solvable quotient

Conjunction with other properties

Conjunction	Other component of conjunction	Additional comments
finite solvable group	finite group	For finite groups, being solvable is equivalent to being polycyclic, and has many other alternative characterizations.
solvable T-group	T-group
solvable HN-group	HN-group

Formalisms

In terms of the group extension operator

This group property can be expressed in terms of the group extension operator and/or group property modifiers that arise from this operator The group property of being solvable can be obtained in either of these equivalent ways:

• By applying the poly operator to the group property of being abelian
• By applying the finite normal series operator to the group property of being abelian
• By applying the finite characteristic series operator to the group property of being abelian

Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.

Testing

The testing problem

Further information: Solvability testing problem

The problem of testing whether a group is solvable or not reduces to the problem of computing its derived series. This can be done when the group is described by means of a generating set, if the normal closure algorithm can be implemented.

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsSolvableGroup
View GAP-testable group properties

To determine whether a group is solvable or not, we cna use the following GAP command:

IsSolvableGroup(group);

where group may be a definition of the group or a name for a group previously defined.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F16

The class 20F16 is used for the general theory of solvable groups, while the class 20D10 (coming under 20D which is for finite groups) focusses on finite solvable groups.

Also closely related is 20F19: Generalizations of nilpotent and solvable groups.

References

Textbook references

Book	Page number	Chapter and section	Contextual information	View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info	105		formal definition
Topics in Algebra by I. N. HersteinMore info	116		formal definition, introduced between exercises
Algebra by Serge Lang, ISBN 038795385XMore info	18		definition in paragraph
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info	121		formal definition
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info	95		definition in paragraph
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444More info	171		definition in paragraph
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907More info	194	Definition 3.4.16	formal definition
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189More info	102	Definition 7.9	formal definition
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716More info	563
Topics in Algebra by I. N. HersteinMore info	116		formal definition, introduced between exercises

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Definition links

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