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
12 - Maximal Subgroups of Type I
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
(12.1) Hypothesis.Let L be a maximal subgroup of G of Type I, H = LF and H′ = [H, H]. Let and let τ be the Dade isometry relative to (A(L), L,G).
(12.2) Assume Hypothesis (12.1).
(a) Let χ ∊ S. There is a subset S(χ) of Irr L such that χ = ΣΨ∊S(χ) φ, and φ(1) is independent of φ ∊ for φ ∊ S(χ). Moreover, τ is defined on Z[∪χ∊SS(χ), L#].
(b) Hypothesis (5.2) holds with the isometry τ of Hypothesis (5.2) being the restriction of τ to Z[S,L#]. If where R1(φ) is an orthonormal subset ofZ[Irr G] of cardinality 2, and
Proof. (a) The first assertion follows from (8.2.c) and (1.7.c). If φ ∊ S (χ), then by (1.5.a), and so H ⊄ Ker φ. By (1.2), it follows that Supp((φ) ⊂ A(L) ∪ {1}, and so τ is defined on.
(b) By (a), τ is defined on Z[S,L#]. If χ ∊ S, then and, by (1.5.e),. The elements of S are pairwise orthogonal by (1.5.c). Let χ ∊ S. By (1.4) applied to S (χ) U S (X), there are pairwise distinct irreducible characters μφ and μ′φ of G and an integer ε = ±1 such that, for φ ∊ S(χ), Then and (5.2.d) holds with.
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- Character Theory for the Odd Order Theorem , pp. 69 - 74Publisher: Cambridge University PressPrint publication year: 2000