Solvable group

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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 \le H_1 \le \ldots \le 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 \le H_1 \le \ldots \le 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 \le H_1 \le \ldots \le 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.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

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

Extreme examples

Groups satisfying the property

Here are some basic/important groups satisfying the property:

 GAP ID
Cyclic group:Z22 (1)
Cyclic group:Z33 (1)
Cyclic group:Z44 (1)
Group of integers
Klein four-group4 (2)
Symmetric group:S36 (1)
Trivial group1 (1)

Here are some relatively less basic/important groups satisfying the property:

 GAP ID
Alternating group:A412 (3)
Dihedral group:D88 (3)
Direct product of Z4 and Z28 (2)
Quaternion group8 (4)
Special linear group:SL(2,3)24 (3)
Symmetric group:S424 (12)

Here are some even more complicated/less basic groups satisfying the property:

 GAP ID
Binary octahedral group48 (28)
Central product of D8 and Z416 (13)
Dihedral group:D1616 (7)
Direct product of A4 and Z224 (13)
Direct product of D8 and Z216 (11)
General linear group:GL(2,3)48 (29)
Generalized quaternion group:Q1616 (9)
M1616 (6)
Mathieu group:M972 (41)
Nontrivial semidirect product of Z4 and Z416 (4)
Semidihedral group:SD1616 (8)

Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:


Here are some relatively less basic/important groups that do not satisfy the property:

 GAP ID
Alternating group:A560 (5)
Alternating group:A6360 (118)
Free group:F2
Projective special linear group:PSL(3,2)168 (42)
Special linear group:SL(2,5)120 (5)
Symmetric group:S5120 (34)
Symmetric group:S6720 (763)

Here are some even more complicated/less basic groups that do not satisfy the property:

 GAP ID
Alternating group:A7
Mathieu group:M10720 (765)
Projective special linear group:PSL(2,11)660 (13)
Projective special linear group:PSL(2,8)504 (156)
Special linear group:SL(2,7)336 (114)
Special linear group:SL(2,9)720 (409)


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

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
pseudovarietal group property Yes solvability is pseudovarietal Solvability is closed under taking subgroups, quotients, and finite direct products (more below).
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.
subgroup-closed group property Yes solvability is subgroup-closed If G is solvable, and H \le 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_1N_2\dots N_r is also solvable.
isoclinism-invariant group property Yes isoclinic groups have same derived length If G_1 and G_2 are isoclinic groups, then G_1 is solvable if and only if G_2 is. Moreover, if so, the derived length of G_1 equals the derived length of G_2, unless one of the groups is trivial and the other is nontrivial abelian.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
abelian group derived subgroup is trivial abelian implies solvable solvable not implies abelian (see also list of examples) Metabelian group, Metanilpotent group, Nilpotent group|FULL LIST, MORE INFO
cyclic group (see also list of examples) Abelian group, Metabelian group, Metacyclic group, Nilpotent group, Polycyclic group|FULL LIST, MORE INFO
nilpotent group lower central series reaches the identity nilpotent implies solvable solvable not implies nilpotent (see also list of examples) Metanilpotent group|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) Polycyclic group|FULL LIST, MORE INFO
polycyclic group subnormal series with cyclic factor groups
equivalent to solvable in the finite case
polycyclic implies solvable solvable not implies polycyclic (see also list of examples) Finitely generated solvable group, Finitely presented solvable group|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;
equivalent to solvable in the finite case
solvable implies hypoabelian hypoabelian not implies solvable Residually solvable group|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
equivalent to solvable in the finite case
residually solvable group every non-identity element has a non-identity image in some solvable quotient
equivalent to solvable in the finite case

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:

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