Character Theory of Finite Groups
Varies from I. M. Isaacs' 1976 book of the same title by updating results, describing new directions the field has taken, being just a bit more module-theoretic, and including many examples in which the character table or at least the character degrees of groups are calculated. Areas in which the earlier work is still current are treated briefly if at all. Most of the material should be accessible to readers with only one term of instruction in group theory. A sampling of topics turns up degrees of irreducible representations, coprime action, Clifford Theory, monomial groups, the degree graph, lengths of conjugacy classes, splitting fields, and three arithmetical applications. Annotation copyrighted by Book News, Inc., Portland, OR
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Examples of groups
Notations and results from group theory
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7t-group abelian group abelian normal subgroup algebraic integer automorphism cd G CG(A CG(g Char character degrees character of G character table characters of degree classes of G conjugacy classes conjugate contradiction cyclic cyclotomic field define dihedral group dimK divides G dl G elements of order extension extraspecial fixed points Frobenius group Frobenius kernel g e G G is p-nilpotent G is solvable G-set group G group of order hence G Hence there exists Huppert implies induction involutions Irr(G irreducible characters Isaacs isomorphic Lemma Let G matrices minimal normal subgroup module monomial nilpotent nonabelian obtain obviously operates fixed-point-freely ord g p-group permutation projective representation Proof proves quaternion group representation of G root of unity shows simple groups simple KG-module skew field solvable group splitting field subgroup of G Suppose G Sylp G Theorem x e Irr G