The proof in G that every CN-group of odd order is solvable requires extensive passages in Chapters 7, 8, and 10 of G. While this material is worthwhile for a deeper understanding of group theory, most of it is unnecessary for the CN-theorem if one combines ideas from Gorenstein's proof with ideas from the proof in W. Feit's Characters of Finite Groups [6]. We indicate how to do this now.

One first reads Chapters 1–8 of G or the substitute prerequisites that are described in Appendix A, as well as Sections 1–4 of this work and Lemma 10.1.3 of G. Then one notes that for G solvable of odd order, Theorems 7.6.1 and 10.2.1 of G follow from our Theorems 4.18(b) and 3.7, while the proofs of Theorems 7.6.2 and 10.3.1 reduce to one paragraph and one sentence respectively. One proceeds to the introduction of Chapter 14 and to Section 14.1, which is slightly easier for G of odd order (but not necessarily solvable). Lemma 14.2.1 is easy, but then it is useful to insert the following lemma, suggested by the proof of (27.6) in [6].

Lemma D.1. Suppose G is a minimal simple CN-group of odd order, p is a prime, P and Q are Sylow p-subgroups of G, and P ∩ Q ≠ 1. Then P = Q.

Proof. Assume the result is false. We will obtain a contradiction. Take P to violate the conclusion for some Q. Let N = NG(Z(J(P))). (One may substitute L(P) for J(P) throughout this proof if one prefers to use Theorem B.4 instead of Theorem 6.2.)