Group of finite composition length: Difference between revisions
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==Definition== | ==Definition== | ||
A group is said to have '''finite composition length''' if it possesses a [[composition series]] of finite length, viz., a [[subnormal series]] | A group is said to have '''finite composition length''' if it satisfies the following equivalent conditions: | ||
# It possesses a [[defining ingredient::composition series]] of finite length, viz., a [[subnormal series]] such that all the successive quotients are [[defining ingredient::simple group]]s. | |||
# It satisfies both the [[defining ingredient::group satisfying ascending chain condition on subnormal subgroups|ascending chain condition on subnormal subgroups]] and the [[defining ingredient::group satisfying descending chain condition on subnormal subgroups|descending chain condition on subnormal subgroups]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Finite group]] | * [[Weaker than::Finite group]] | ||
===Weaker properties=== | |||
* [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] | |||
* [[Stronger than::Group satisfying descending chain condition on subnormal subgroups]] | |||
* [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] | |||
* [[Stronger than::Group satisfying descending chain condition on normal subgroups]] | |||
Revision as of 19:20, 14 November 2008
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 is a variation of finiteness (groups)|Find other variations of finiteness (groups) |
This property makes sense for infinite groups. For finite groups, it is always true
Definition
A group is said to have finite composition length if it satisfies the following equivalent conditions:
- It possesses a composition series of finite length, viz., a subnormal series such that all the successive quotients are simple groups.
- It satisfies both the ascending chain condition on subnormal subgroups and the descending chain condition on subnormal subgroups.