Solvable group: Difference between revisions
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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. | 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. | ||
==Definition links== | |||
* [http://en.wikipedia.org/wiki/Solvable_group Wikipedia definition] | |||
* [http://planetmath.org/encyclopedia/SolvableGroup.html Planetmath definition] | |||
* [http://mathworld.wolfram.com/SolvableGroup.html Mathworld definition] | |||
Revision as of 16:50, 26 March 2007
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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
Origin
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.
Definition
Symbol-free definition
A group is said to be solvable if any of the following equivalent conditions holds:
- There is a normal series starting from the trivial subgroup and ending at the whole group with each successive quotient being an Abelian group
- There is a strongly normal series 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.
Definition with symbols
Relation with other properties
Stronger properties
Weaker properties
Conjunction with other properties
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.