Difference between revisions of "Fitting-free group"

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===Symbol-free definition===
 
===Symbol-free definition===
  
A [[subgroup]] of 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''' if it satisfies the following equivalent conditions:
  
 
* It has no nontrivial [[nilpotent normal subgroup]]
 
* It has no nontrivial [[nilpotent normal subgroup]]
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* it has no nontrivial [[Abelian normal subgroup]]
 
* Its [[Fitting subgroup]] is trivial
 
* Its [[Fitting subgroup]] is trivial
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===Equivalence of definitions===
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 +
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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Clearly, since every [[Abelian group]] is [[nilpotent group|nilpotent]], the first condition implies the second.
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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).
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<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>.
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Also <math>Z(N)</math> is clearly Abelian.
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Thus, starting from a nontrivial nilpotent normal subgroup, we have obtained a nontrivial Abelian normal subgroup.
  
 
==Relation with other properties==
 
==Relation with other properties==
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===Weaker properties===
 
===Weaker properties===
  
* [[NA-free group]]
 
 
* [[Centerless group]]
 
* [[Centerless group]]

Revision as of 06:52, 26 March 2007

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
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Definition

Symbol-free definition

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

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.

Clearly, since every Abelian group is nilpotent, the first condition implies the second.

To show that the reverse implication, suppose there is a nontrivial nilpotent normal subgroup N in G. Then, Z(N) (viz the center of N) is nontrivial (this follows from the definition of nilpotence).

Z(N) is a characteristic subgroup inside N and N is normal inside G. Since a characteristic subgroup of a normal subgroup is normal, we conclude that Z(N) \triangleleft G.

Also Z(N) is clearly Abelian.

Thus, starting from a nontrivial nilpotent normal subgroup, we have obtained a nontrivial Abelian normal subgroup.

Relation with other properties

Stronger properties

Weaker properties