Representations of Solvable Groups
Representation 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.
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1r-group 1r-number absolutely irreducible acts faithfully acts irreducibly argue by induction assume that G bijection Brauer characters CG(v CG(Z characteristic subgroup chief factor choose completely reducible conjugacy class conjugate contradiction Corollary 1.10 defect group deﬁne deﬁnition dim(V dl(G elementary abelian exists extends factor of G faithful irreducible ﬁeld ﬁnite ﬁrst ﬁxed ﬁxes follows Frobenius group Furthermore G 1r G be solvable G-invariant group G Hall 1r-subgroup Hence homogeneous components IG(a integer Irr G irreducible character irreducible constituent irreducible G-module isomorphic last paragraph Lemma Let G linear groups minimal normal subgroup module nilpotent non-abelian non-trivial P G Syl,,(G p-block p-group p-rational p-regular permutation group prime q primitive permutation primitive permutation group Proposition prove quasi-primitive regular orbit satisﬁes solvable group Step subgroup of G Sylow p-subgroup Theorem 9.3 transitively permutes vector space X G Irr X G Irr(G yields Zsigmondy prime divisor