Solvable group

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>

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
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

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 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 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 solvable length of the solvable group.

Formalisms

In terms of group properties

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

Relation with other properties

Stronger properties

• Nilpotent group
• Supersolvable group
• Polycyclic group

Weaker properties

• HypoAbelian group
• Imperfect group

Conjunction with other properties

• Solvable T-group
• Solvable HN-group

Metaproperties

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 the fact that the derived series of a subgroup is contained, term-wise, in the derived series of the whole group.

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 the fact that the derived series of a quotient group is term-wise the quotient of the derived series of the group.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A finite direct product of solvable groups is solvable. This follows from the fact that the derived series of a direct product is the direct product of the derived series of the individual subgroups.

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).
View GAP-testable group properties

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

IsSolvable (group);

where

group

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

Definition links

• Wikipedia page
• Planetmath page
• Mathworld page