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 $\{ e \} = H_0 \le H_1 \le \ldots \le H_n = G$ 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

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)

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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
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
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

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.