Among the main tools for shortening the first half of the proof of FT are Theorems 6.1 and 6.2, which are obtained by use of the concept of p-stability. In Section 6 these are obtained from theorems in G, which have shorter proofs if one restricts to groups of odd order and uses a different characteristic subgroup in place of J(S). In this appendix and Appendix B we outline these shorter proofs. Although we use some results from Chapters 1–6 of G, this makes it unnecessary to use some other results from G, as described below.

This appendix is devoted mainly to proving Theorem 6.1 and a special case of Theorem 6.5.3 of G that will be applied in Appendix B. For those who wish to read both this appendix and Appendix B, the prerequisites for this book may be reduced and handled as follows. One first reads Chapters 1–6 and Section 7.3 of G, except for Theorems 2.8.3 and 2.8.4 (pp. 42–55) and Sections 3.8 and 6.5. Next one reads Sections 1 and 2 in Chapter I of this book, followed by this appendix and Appendix B (including parts of Sections 3.8 and 6.5 of G mentioned later in this appendix). In particular, one does not need to read Chapter 8 and most of Chapter 7 of G.

Additional prerequisites for the proof of the CN-theorem are described in Appendix D.

To begin, we refer the reader to pages 39–40 of G, which introduce the groups GL(2, q) and several related families of groups.