Difference between revisions of "Fitting-free group"

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==Definition==
 
==Definition==
  
 
===Symbol-free definition===
 
===Symbol-free definition===
  
A [[group]] is said to be '''Fitting-free''' if it satisfies the following equivalent conditions:
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A [[group]] is said to be '''Fitting-free''' (or sometimes, '''semisimple''') if it satisfies the following equivalent conditions:
  
* It has no nontrivial [[nilpotent normal subgroup]]
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# It has no nontrivial [[defining ingredient::abelian normal subgroup]].
* it has no nontrivial [[Abelian normal subgroup]]
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# It has no nontrivial [[defining ingredient::nilpotent normal subgroup]].
* Its [[Fitting subgroup]] is trivial
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# It has no nontrivial [[defining ingredient::solvable normal subgroup]].
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# Its [[defining ingredient::Fitting subgroup]] is trivial.
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# Its [[defining ingredient::solvable radical]] is trivial.
  
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When the group is finite, this is equivalent to the following:
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* It has no nontrivial [[defining ingredient::abelian characteristic subgroup]]
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* It has no nontrivial [[defining ingredient::nilpotent characteristic subgroup]]
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* It has no nontrivial [[defining ingredient::solvable characteristic subgroup]]
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* It has no nontrivial [[defining ingredient::elementary abelian normal subgroup]]
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* It has no nontrivial [[defining ingredient::elementary abelian characteristic subgroup]]
 
===Equivalence of definitions===
 
===Equivalence of definitions===
  
We want to show that the condition of having no nontrivial nilpotent normal subgroup is equivalent to the condition of having non nontrivial Abelian normal subgroup.
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The key idea is to use the fact that any nontrivial solvable group has a nontrivial [[abelian characteristic subgroup]]. {{further|[[Equivalence of definitions of Fitting-free group]]}}
 
 
Clearly, since every [[Abelian group]] is [[nilpotent group|nilpotent]], the first condition implies the second.
 
 
 
To show that the reverse implication, suppose there is a nontrivial nilpotent normal subgroup <math>N</math> in <math>G</math>. Then, <math>Z(N)</math> (viz the center of <math>N</math>) is nontrivial (this follows from the definition of nilpotence).
 
 
 
<math>Z(N)</math> is a [[characteristic subgroup]] inside <math>N</math> and <math>N</math> is [[normal subgroup|normal]] inside <math>G</math>. Since a [[characteristic of normal implies normal|characteristic subgroup of a normal subgroup is normal]], we conclude that <math>Z(N) \triangleleft G</math>.
 
 
 
Also <math>Z(N)</math> is clearly Abelian.
 
 
 
Thus, starting from a nontrivial nilpotent normal subgroup, we have obtained a nontrivial Abelian normal subgroup.
 
  
 
==Relation with other properties==
 
==Relation with other properties==

Latest revision as of 22:29, 6 August 2009

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
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 Fitting-free (or sometimes, semisimple) if it satisfies the following equivalent conditions:

  1. It has no nontrivial abelian normal subgroup.
  2. It has no nontrivial nilpotent normal subgroup.
  3. It has no nontrivial solvable normal subgroup.
  4. Its Fitting subgroup is trivial.
  5. Its solvable radical is trivial.

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