Monolithic group: Difference between revisions
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[[Importance rank::3| ]] | |||
{{group property}} | {{group property}} | ||
{{variationof|simplicity}} | {{variationof|simplicity}} | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is said to be '''monolithic''' if it has a unique [[minimal normal subgroup]]. This minimal normal subgroup is termed a '''monolith'''. | A [[group]] is said to be '''monolithic''' if it has a unique [[defining ingredient::minimal normal subgroup]], and this is contained in ''every'' nontrivial [[defining ingredient::normal subgroup]]. This minimal normal subgroup is termed a '''monolith'''. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Simple group]] | * [[Weaker than::Simple group]] | ||
* [[ | |||
===Weaker properties=== | |||
* [[Stronger than::Subdirectly irreducible group]] | |||
* [[Stronger than::Directly indecomposable group]] | |||
===Related properties=== | ===Related properties=== | ||
* [[One-headed group]] | * [[One-headed group]] | ||
Latest revision as of 03:34, 17 December 2011
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 simplicity|Find other variations of simplicity | Read a survey article on varying simplicity
Definition
Symbol-free definition
A group is said to be monolithic if it has a unique minimal normal subgroup, and this is contained in every nontrivial normal subgroup. This minimal normal subgroup is termed a monolith.