Book contents
- Frontmatter
- Cotents
- Preface
- Part I Character Theory for the Odd Order Theorem
- Part II A Theorem of Suzuki
- Introduction
- Notation
- Chapter I General Properties of G
- Chapter II The First Case
- Chapter III The Structure of H
- Chapter IV Characterization of PSU(3, q)
- 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
Chapter II - The First Case
Published online by Cambridge University Press: 05 September 2013
- Frontmatter
- Cotents
- Preface
- Part I Character Theory for the Odd Order Theorem
- Part II A Theorem of Suzuki
- Introduction
- Notation
- Chapter I General Properties of G
- Chapter II The First Case
- Chapter III The Structure of H
- Chapter IV Characterization of PSU(3, q)
- 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
For the proof of the main theorem of this chapter, the Hall-Wieland t Theorem (see [Ha], Theorem 14.4.2) is needed. Let G be a finite group, p a prime number and P a Sylow p-subgroup of G. A subgroup A of P is said to be weakly closed in P relative to G if, for all g ϵ G, Ag ⊂ P implies that Ag = A. Let Zp−1(P) be the (p − 1)st term of the upper central series of P. Let A be a weakly closed subgroup of P relative to G. The Hall-Wielandt Theorem states that, if A ⊂ Zp−1(P), or if p > 2 and A is abelian, then G/Op(G) is isomorphic to NG(A)/Op(NG(A)).
In this chapter it will be assumed that
(Bl) The subgroup V contains a subgroup P of prime order p such that C G (P) has 2-rank 1.
We will demonstrate the following theorem.
Theorem B.The conclusion of Theorem A holds for G under the hypothesis (Bl).
If G has a normal subgroup of index p, the conclusion of the theorem holds by Chapter I, § 3, Proposition 2. It will be assumed therefore that
(B2)G has no normal subgroup of index p.
- Type
- Chapter
- Information
- Character Theory for the Odd Order Theorem , pp. 108 - 114Publisher: Cambridge University PressPrint publication year: 2000