Difference between revisions of "Fitting-free group"

From Groupprops
Jump to: navigation, search
Line 1: Line 1:
 +
{{stdnonbasicdef}}
 
{{group property}}
 
{{group property}}
 
 
{{variationof|simplicity}}
 
{{variationof|simplicity}}
 
{{stdnonbasicdef}}
 
 
 
==Definition==
 
==Definition==
  

Revision as of 12:41, 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

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