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.
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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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Symbol-free definition

A group is said to be Fitting-free if it satisfies the following equivalent conditions:

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.

Clearly, since every Abelian group is nilpotent, the first condition implies the second.

To show that the reverse implication, suppose there is a nontrivial nilpotent normal subgroup N in G. Then, Z(N) (viz the center of N) is nontrivial (this follows from the definition of nilpotence).

Z(N) is a characteristic subgroup inside N and N is normal inside G. Since a characteristic subgroup of a normal subgroup is normal, we conclude that Z(N) \triangleleft G.

Also Z(N) is clearly Abelian.

Thus, starting from a nontrivial nilpotent normal subgroup, we have obtained a nontrivial Abelian normal subgroup.

Relation with other properties

Stronger properties

Weaker properties