# Difference between revisions of "Fitting-free group"

From Groupprops

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* 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 [[solvable normal subgroup]] | ||

* Its [[Fitting subgroup]] is trivial | * Its [[Fitting subgroup]] is trivial | ||

===Equivalence of definitions=== | ===Equivalence of definitions=== | ||

− | + | 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]]}} | |

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==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: (factscloselyrelated 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 propertiesVIEW 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:

- It has no nontrivial nilpotent normal subgroup
- It has no nontrivial Abelian normal subgroup
- It has no nontrivial solvable normal subgroup
- Its Fitting subgroup is trivial

### Equivalence of definitions

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