Difference between revisions of "Fitting-free group"

From Groupprops
Jump to: navigation, search
Line 9: Line 9:
  
 
* It has no nontrivial [[nilpotent normal subgroup]]
 
* It has no nontrivial [[nilpotent normal subgroup]]
* it has no nontrivial [[Abelian normal subgroup]]
+
* It has no nontrivial [[Abelian normal subgroup]]
 +
* It has no nontrivial [[solvable normal subgroup]]
 
* Its [[Fitting subgroup]] is trivial
 
* Its [[Fitting subgroup]] is trivial
  
 
===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.
+
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==

Revision as of 12:42, 11 March 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
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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
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 Fitting-free if it satisfies the following equivalent conditions:

Equivalence of definitions

The key idea is to use the fact that any nontrivial solvable group has a nontrivial Abelian characteristic subgroup. Template:FurtheR

Relation with other properties

Stronger properties

Weaker properties