# Fitting-free group

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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

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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## 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`