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
5 - Coherence
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
(5.1) Definition. Let L and G be finite groups, A ⊂ L and S ⊂ Z[Irr L]. Let τ be a Z-linear isometry from E to Z[Irr G], where E is a Z-module such that Z[S,A] ⊂ E ⊂ Z[Irr L]. We say that (S, A, τ) is coherent, or that S is coherent, if Z[S, A] ≠ 0 and if there is a linear isometry from Z[S] to Z[Irr G] which coincides with τ on Z[S, A].
(5.2) Hypothesis, (a) Let L and G be finite groups and let S be a non-empty set of characters of L. Assume that, if χ ∈ S, then and.
(b) Assume that τ is a linear isometry fromZ[S, L#] toZ[Irr G, G#].
(c) The elements of S are pairwise orthogonal.
(d) Assume that, for for some orthonormal subset R(χ) ofZ[Irr G].
(e) If χ ∈ S, ϕ ∈ S and ϕ is orthogonal to then R(ϕ) is orthogonal to R(χ).
(5.3) (a) Assume (5.2.a), (5.2.b) and that S ⊂ Irr L. Then Hypothesis (5.2) holds.
(b) Assume Hypothesis (4.6), (5.2.a) and that
Then Hypothesis (5.2) holds with the isometry τ of Hypothesis (5.2) being the restriction toZ[S, L#] of the isometry τ of Hypothesis (4.6).
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- Character Theory for the Odd Order Theorem , pp. 25 - 29Publisher: Cambridge University PressPrint publication year: 2000