# Fitting-free group

From Groupprops

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