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 if it satisfies the following equivalent conditions:
- It has no nontrivial nilpotent normal subgroup
- it has no nontrivial Abelian normal subgroup
- Its Fitting subgroup is trivial
Equivalence of definitions
We want to show that the condition of having no nontrivial nilpotent normal subgroup is equivalent to the condition of having non nontrivial Abelian normal subgroup.
To show that the reverse implication, suppose there is a nontrivial nilpotent normal subgroup in . Then, (viz the center of ) is nontrivial (this follows from the definition of nilpotence).
is a characteristic subgroup inside and is normal inside . Since a characteristic subgroup of a normal subgroup is normal, we conclude that .
Also is clearly Abelian.
Thus, starting from a nontrivial nilpotent normal subgroup, we have obtained a nontrivial Abelian normal subgroup.