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.
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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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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