Classification of finite solvable CN-groups

This article gives the statement, and possibly proof, of a statement where the conclusion of the statement involves a disjunction (OR) of two possibilities. The prototypical form is: "every A is a B or a C."

Statement

Suppose $G$ is a finite group that is both a finite solvable group and a CN-group (hence also a finite CN-group. Then, $G$ must be of one of these three types:

1. $G$ is a finite nilpotent group.
2. $G$ is a Frobenius group where the Frobenius complement is a finite group in which every abelian subgroup is cyclic.
3. $G$ is a 3-step group.

Proof

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References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 402, Theorem 14.1.5, ^{More info}