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Let G be a finite group and let A be a finite solvable operator group on G. Suppose that A and G have relatively prime orders. Let T be the fixed-point subgroup of G with respect to A. We say that A fixes a complex character ζ of G if ζ (gα) = ζ (g) for all g ∈ G and α ϵ A. Our aim in this paper is to define a one-to-one correspondence between the irreducible characters of T and those irreducible characters of G that are fixed by A, and to prove some properties of this correspondence that were mentioned in (8).
Let p be a prime, and let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:
(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.
In this paper a simplified proof of a theorem of Sanov (4) is given. No mention is required of Lie elements or of the Baker-Hausdorff formula, both of which played central roles in Sanov's proof.
Many problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.
About 30 years ago, Walter Feit and John G. Thompson [8] proved the Odd Order Theorem, which states that all finite groups of odd order are solvable. In the words of Daniel Gorenstein [15, p. 14], “it is not possible to overemphasize the importance of the Feit-Thompson Theorem for simple group theory.” Their proof consists of a set of preliminary results followed by three parts-local analysis, characters, and generators and relations- corresponding to Chapters IV, V, and VI of their paper (denoted by FT here). Local analysis of a finite group G means the study of the structure of, and the interaction between, the centralizers and normalizers of nonidentity p-subgroups of G. Here Sylow's Theorem is the first main tool. The main purpose of this book is to present a new version of the local analysis of a minimal counterexample G to the Feit-Thompson Theorem, that is, of Chapter IV and its preliminaries. We also include a remarkably short and elegant revision of Chapter VI by Thomas Peterfalvi in Appendix C.
What we would ideally like to prove, but cannot, is that each maximal subgroup M of G has a nonidentity proper normal subgroup M0 such that
(1) CM0(α) = 1, for all elements α ∈ M − M0,
(2) for all elements g ∈ G ∈ M,
(3) M0 is nilpotent,
(4) M/M0 is cyclic,
and such that the totality of these subgroups M0, with M ranging over all of the maximal subgroups of G, forms a partition of G:
(5) each nonidentity element of G lies in exactly one of the subgroups M0.
As mentioned in the preface, the proof of the Feit-Thompson Theorem is similar in broad outline to the proof for the special case of CN-groups. There, the CN-group hypothesis yields immediately that the maximal subgroups are Frobenius groups or “three step groups” under a definition different from our definition of three step groups (given in a remark before Proposition 16.1). In contrast, here we have no preliminary restrictions on a maximal subgroup M other than its being solvable. However, having proved the Uniqueness Theorem, we are able to show in fairly short order that M has p-length one for every prime p (Theorem 10.6). Eventually we show that either M is “of Frobenius type” (“almost” a Frobenius group), or M is a three step group (as defined in these notes) (Theorem I). (Incidently, we can obtain Burnside's paqb theorem for odd primes p and q very easily now, as shown in the remark after the proof of Theorem 10.2).
In this chapter we attain part of our final goal by focusing our attention on a single maximal subgroup M. We introduce two normal Hall subgroups Mα and Mσ of M and study their properties in Section 10. The subgroup Mσ plays a role analogous to that of the Fitting subgroup (i.e., the Frobenius kernel) in a Frobenius group. Indeed, if M is a Frobenius group, then Mσ = F(M) and r(M/Mσ) = 1. In Section 11 we study the structure of M under a particular restriction.
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.)
In 1963 Walter Feit and John G. Thompson published a proof of a 1911 conjecture by Burnside that every finite group of odd order is solvable. This proof, which ran for 255 pages, was a tour-de-force of mathematics and inspired intense effort to classify finite simple groups. This book presents a revision and expansion of the first half of the proof of the Feit–Thompson theorem. Simpler, more detailed proofs are provided for some intermediate theorems. Recent results are used to shorten other proofs. The book will make the first half of this remarkable proof accessible to readers familiar with just the rudiments of group theory.
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.