Difference between revisions of "Fitting-free group"

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A [[group]] is said to be '''Fitting-free''' if it satisfies the following equivalent conditions:
 
A [[group]] is said to be '''Fitting-free''' if it satisfies the following equivalent conditions:
  
* It has no nontrivial [[nilpotent normal subgroup]]
+
* It has no nontrivial [[defining ingredient::nilpotent normal subgroup]]
* It has no nontrivial [[Abelian normal subgroup]]
+
* It has no nontrivial [[defining ingredient::Abelian normal subgroup]]
* It has no nontrivial [[solvable normal subgroup]]
+
* It has no nontrivial [[defining ingredient::solvable normal subgroup]]
* Its [[Fitting subgroup]] is trivial
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* Its [[defining ingredient::Fitting subgroup]] is trivial
  
 +
When the group is finite, this is equivalent to the following:
 +
 +
* It has no nontrivial [[defining ingredient::nilpotent characteristic subgroup]]
 +
* It has no nontrivial [[defining ingredient::Abelian characteristic subgroup]]
 +
* It has no nontrivial [[defining ingredient::solvable characteristic subgroup]]
 +
* It has no nontrivial [[defining ingredient::elementary Abelian normal subgroup]]
 +
* It has no nontrivial [[defining ingredient::elementary Abelian characteristic subgroup]]
 
===Equivalence of definitions===
 
===Equivalence of definitions===
  

Revision as of 16:10, 19 May 2008

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to Fitting-free group, all facts related to Fitting-free group) |Survey articles about this | Survey articles about definitions built on this
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View a list of other standard non-basic definitions
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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Definition

Symbol-free definition

A group is said to be Fitting-free if it satisfies the following equivalent conditions:

When the group is finite, this is equivalent to the following:

Equivalence of definitions

The key idea is to use the fact that any nontrivial solvable group has a nontrivial Abelian characteristic subgroup. Further information: Equivalence of definitions of Fitting-free group

Relation with other properties

Stronger properties

Weaker properties