# Difference between revisions of "Fitting-free group"

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+ | {{stdnonbasicdef}} | ||

{{group property}} | {{group property}} | ||

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{{variationof|simplicity}} | {{variationof|simplicity}} | ||

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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: | + | A [[group]] is said to be '''Fitting-free''' (or sometimes, '''semisimple''') if it satisfies the following equivalent conditions: |

− | + | # It has no nontrivial [[defining ingredient::abelian normal subgroup]]. | |

− | + | # It has no nontrivial [[defining ingredient::nilpotent normal subgroup]]. | |

− | + | # It has no nontrivial [[defining ingredient::solvable normal subgroup]]. | |

+ | # Its [[defining ingredient::Fitting subgroup]] is trivial. | ||

+ | # Its [[defining ingredient::solvable radical]] is trivial. | ||

+ | When the group is finite, this is equivalent to the following: | ||

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+ | * It has no nontrivial [[defining ingredient::abelian characteristic subgroup]] | ||

+ | * It has no nontrivial [[defining ingredient::nilpotent characteristic subgroup]] | ||

+ | * It has no nontrivial [[defining ingredient::solvable characteristic subgroup]] | ||

+ | * It has no nontrivial [[defining ingredient::elementary abelian normal subgroup]] | ||

+ | * It has no nontrivial [[defining ingredient::elementary abelian characteristic subgroup]] | ||

===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== |

## 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: (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** (or sometimes, **semisimple**) if it satisfies the following equivalent conditions:

- It has no nontrivial abelian normal subgroup.
- It has no nontrivial nilpotent normal subgroup.
- It has no nontrivial solvable normal subgroup.
- Its Fitting subgroup is trivial.
- Its solvable radical is trivial.

When the group is finite, this is equivalent to the following:

- It has no nontrivial abelian characteristic subgroup
- It has no nontrivial nilpotent characteristic subgroup
- It has no nontrivial solvable characteristic subgroup
- It has no nontrivial elementary abelian normal subgroup
- It has no nontrivial elementary abelian characteristic subgroup

### 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`