Representations of Solvable GroupsRepresentation theory plays an important role in algebra, and in this book Manz and Wolf concentrate on that part of the theory which relates to solvable groups. The authors begin by studying modules over finite fields, which arise naturally as chief factors of solvable groups. The information obtained can then be applied to infinite modules, and in particular to character theory (ordinary and Brauer) of solvable groups. The authors include proofs of Brauer's height zero conjecture and the Alperin-McKay conjecture for solvable groups. Gluck's permutation lemma and Huppert's classification of solvable two-transive permutation groups, which are essentially results about finite modules of finite groups, play important roles in the applications and a new proof is given of the latter. Researchers into group theory, representation theory, or both, will find that this book has much to offer. |
Contents
Chap 0 Preliminaries | 1 |
Chap I Solvable subgroups of linear groups | 27 |
Chap II Solvable permutation groups | 73 |
Chap Ill Module actions with large centralizers | 117 |
Chap IV Prime power divisors of character degrees | 157 |
Chap V Complexity of character degrees | 210 |
Chap VI πspecial characters | 265 |
| 293 | |
| 299 | |
Common terms and phrases
absolutely irreducible acts faithfully acts irreducibly argue by induction assume that G bijection Brauer characters centralizes CG(v CG(x CG(Z char chief factor choose Clifford's Theorem completely reducible Consequently contradiction Corollary 1.10 Cv(P defect group dim(V dl(G elementary abelian exists extends factor of G faithful irreducible finite follows Frobenius group Furthermore G be solvable G-invariant G/F-module group G Hence homogeneous components induction on G integer Irr G Irr G|0 irreducible character irreducible constituent irreducible G-module isomorphic last paragraph Lemma Let G linear groups minimal normal subgroup module nilpotent non-abelian non-trivial Op(G p-block p-group p-rational p-regular p-solvable permutation group prime q primitive permutation primitive permutation group Proposition prove quasi-primitive regular orbit solvable group Step subgroup of G suffices to show Sylow p-subgroup Sylp(G Theorem 9.3 transitively permutes V₁ vector space W₁ yields Zsigmondy prime divisor µ Є