Solvable group

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
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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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

History

This term was introduced by: Galois

The notion of solvable group arose from the attempt to characterize the Galois groups of those field extensions which could be solved by means of radicals.

In fact, the term solvable arose precisely because a normal field extension is contained in a radical extension if and only if the Galois group is solvable.

Definition

Symbol-free definition

A group is said to be solvable (or soluble) if any of the following equivalent conditions holds:

• There is a normal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient being an abelian group.
• There is a subnormal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient being an abelian group.
• The derived series reaches the identity in finitely many steps. The number of steps needed is termed the derived length (sometimes also called solvable length) of the solvable group.
• There is a characteristic series of finite length with each successive quotient being an abelian group.
• There is a fully invariant series of finite length with each successive quotient being an abelian group.

Definition with symbols

A group $G$ is said to be solvable if it satisfies any of the following equivalent conditions;

• 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 $H_{i}/H_{i-1}$ is Abelian.

• There exists a series of subgroups:

$e=H_{0}\triangleleft H_{1}\triangleleft \ldots \triangleleft H_{n}=G$

such that each $H_{i-1}$ is normal in $H_{i}$ and each $H_{i}/H_{i-1}$ is Abelian.

• The derived series of $G$, viz the series $G^{(n)}$ where $G^{(0)}=G$ and $G^{(i+1)}=[G^{(i)},G^{(i)}]$, reaches the trivial subgroup in finitely many steps.

Equivalence of definitions

Further information: Equivalence of definitions of solvable 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.

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.

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)	For intermediate notions between solvable group and nilpotent group, click here.	nilpotent versus solvable
metabelian group	abelian normal subgroup with abelian quotient; derived length two		(see also list of examples)	For intermediate notions between metabelian group and solvable group, click here.
supersolvable group	normal series with cyclic factor groups	supersolvable implies solvable	solvable not implies supersolvable (see also list of examples)	For intermediate notions between solvable group and supersolvable group, click here.
polycyclic group	subnormal series with cyclic factor groups	polycyclic implies solvable	solvable not implies polycyclic (see also list of examples)	For intermediate notions between solvable group and polycyclic group, click here.
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

Metaproperties

The group property of being solvable is idempotent with respect to the group extension operator. In other words, if a group has a solvable normal subgroup, and the quotient group is solvable as an abstract group, then the whole group is solvable.

In fact, we can just take a subnormal series corresponding to the normal subgroup and pull back a subnormal series corresponding to the quotient group, and put the two subnormal series together to obtain a subnormal series for the whole group.

Quasivarietal group property

This group property is varietal, in the sense that the collection of groups satisfying this property forms a quasivariety of algebras. In other words, the collection of groups satisfying this property is closed under taking subgroups, taking quotients and taking finite direct products
View other quasivarietal group properties

The property of being solvable of solvable length at most $d$, is a varietal group property -- it is in fact equationally defined by the vanishing of the commutator of any $d$ elements. From this, we can deduce that the group property of being solvable is quasivarietal:

• Any subgroup of a solvable group is solvable, in fact, with the same (or smaller) solvable length
• Any quotient of a solvable group is solvable, in fact, with the same (or smaller) solvable length
• Any finite direct product of solvable groups is solvable, in fact, with solvable length bounded by the maximum of the solvable lengths of the groups

For full proof, refer: Solvability is quasivarietal

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a solvable group is solvable. This follows from its being quasivarietal. See above.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a solvable group is solvable. This follows from its being quasivarietal. See above.

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of solvable groups is solvable. This follows from its being quasivarietal. See above.

Finite normal joins

This group property is finite normal join-closed: in other words, a join of finitely many normal subgroups each having the group property, also has the group property

A join of finitely many solvable normal subgroups is also solvable.

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

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

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Definition links

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