Book contents
- Frontmatter
- Cotents
- Preface
- Part I Character Theory for the Odd Order Theorem
- Introduction
- Notation
- 1 Preliminary Results from Character Theory
- 2 The Dade Isometry
- 3 TI-Subsets with Cyclic Normalizers
- 4 The Dade Isometry for a Certain Type of Subgroup
- 5 Coherence
- 6 Some Coherence Theorems
- 7 Non-existence of a Certain Type of Group of Odd Order
- 8 Structure of a Minimal Simple Group of Odd Order
- 9 On the Maximal Subgroups of G of Types II, III and IV
- 10 Maximal Subgroups of Types III, IV and V
- 11 Maximal Subgroups of Types III and IV
- 12 Maximal Subgroups of Type I
- 13 The Subgroups S and T
- 14 Non-existence of G
- Notes
- References
- Part II A Theorem of Suzuki
- Appendix I A Special Case of a Theorem of Huppert
- Appendix II On Near-Fields
- Appendix III On Suzuki 2-Groups
- Appendix IV The Feit-Sibley Theorem
- References
- Index to Parts I and II
11 - Maximal Subgroups of Types III and IV
Published online by Cambridge University Press: 05 September 2013
- Frontmatter
- Cotents
- Preface
- Part I Character Theory for the Odd Order Theorem
- Introduction
- Notation
- 1 Preliminary Results from Character Theory
- 2 The Dade Isometry
- 3 TI-Subsets with Cyclic Normalizers
- 4 The Dade Isometry for a Certain Type of Subgroup
- 5 Coherence
- 6 Some Coherence Theorems
- 7 Non-existence of a Certain Type of Group of Odd Order
- 8 Structure of a Minimal Simple Group of Odd Order
- 9 On the Maximal Subgroups of G of Types II, III and IV
- 10 Maximal Subgroups of Types III, IV and V
- 11 Maximal Subgroups of Types III and IV
- 12 Maximal Subgroups of Type I
- 13 The Subgroups S and T
- 14 Non-existence of G
- Notes
- References
- Part II A Theorem of Suzuki
- Appendix I A Special Case of a Theorem of Huppert
- Appendix II On Near-Fields
- Appendix III On Suzuki 2-Groups
- Appendix IV The Feit-Sibley Theorem
- References
- Index to Parts I and II
Summary
(11.1) Let P and q be odd prime numbers, p ≠ q. Then pq > 4q2 + 1.
Proof. Suppose that q = 3. Then pq > 35 ≥ 37 = 4q2 + 1. Suppose that q > 5. Since p ≥ 3, it suffices to show that 3x > 4x2 + 1 for any integer x ≥ 5. First of all, 35 = 243 > 101 = 4(52) +1. Suppose that x ≥ 5 and that 3x > 4x2 + 1. Then 3x + 1 > 3(4x2 + 1). But 3(4x2 + 1) − 4(x + 1)2 − 1 = 8x2 − 8x − 2 ≥ 0, and so 3x+1 > 4(x + 1)2 + 1.
(11.2) Hypothesis.Assume Hypothesis (10.1) with M of Type III or IV. We assume that H and U have the same meaning as in Definition (8.4) and that H′ - [H,H], U′ = [U,U] and C = CU (H). Let H0 be a normal subgroup of M contained in H which satisfies (9.4). Set p = ∣W2∣ and q = ∣W1∣. For X ⊂ M, S(X) = {χ ∊ S ∣ X ⊂ Kerχ}.
We remark that the set which is denoted by S in Hypothesis (9.5) is denoted by S − S(H) here.
(11.3) Assume Hypothesis (11.2). Then S(H0C) is not coherent.
Proof. Suppose that S(H0C) is coherent. We show that the hypotheses of Theorem (6.3) hold with the symbols L, K, M, H and H1 of Theorem (6.3) replaced by M, M′, 1, HC and H0C respectively. Statement (6.3.a) holds because HC is nilpotent, (6.3.b) comes from the hypothesis on S(H0C) and (6.3.c) follows from (9.6) and (11.1). By Theorem (6.3), S is coherent, which contradicts Theorem (10.8).
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- Character Theory for the Odd Order Theorem , pp. 64 - 68Publisher: Cambridge University PressPrint publication year: 2000