Minimax group: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is said to be a '''minimax group''' if it has a [[subnormal series]] of finite length such that each successive quotient satisfies either the minimum condition or the maximum condition on subgroups. | A [[group]] is said to be a '''minimax group''' if it has a [[subnormal series]] of finite length such that each successive quotient satisfies either the minimum condition on subgroups (i.e., is a [[defining ingredient::slender group]]) or the maximum condition on subgroups (i.e., is a [[defining ingredient::Artinian group]]). | ||
===Definition with symbols=== | ===Definition with symbols=== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[ | * [[Weaker than::Finite group]] | ||
* [[Artinian group]] | * [[Weaker than::Slender group]] | ||
* [[Weaker than::Artinian group]] | |||
===Conjunction with other properties=== | ===Conjunction with other properties=== | ||
* [[Solvable minimax group]] is the conjunction with the property of being a [[solvable group]] | * [[Solvable minimax group]] is the conjunction with the property of being a [[solvable group]] | ||
Revision as of 23:13, 1 March 2009
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 property makes sense for infinite groups. For finite groups, it is always true
Definition
Symbol-free definition
A group is said to be a minimax group if it has a subnormal series of finite length such that each successive quotient satisfies either the minimum condition on subgroups (i.e., is a slender group) or the maximum condition on subgroups (i.e., is a Artinian group).
Definition with symbols
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Relation with other properties
Stronger properties
Conjunction with other properties
- Solvable minimax group is the conjunction with the property of being a solvable group