Pure Mathematics


Representation theory of groups


Péter Hermann, Péter Pál Pálfy, József Pelikán






Representations, modules over group algebras; Wedderburn’s theorem on semisimple algebras; Maschke’s theorem; the number of irreducible representations (over algebraically closed fields of characteristic 0). Characters. Central idempotents, the orthogonality relations. Characterizing equivalence of representations. Decomposition of the regular character. Character tables. Operations for characters: sums, products, algebraic conjugate, determinant. Permutation character. Powers of faithful characters contain all irreducible characters as constituents.

c(g) and c(g)|G: CG(g)| / c(1) are algebraic integers. Burnside’s theorem: G is solvable if |G| is the product of two prime powers. c(1)ï |G:Z(G)|.

Induced characters. The reciprocity law. The permutation character as an induced character. Monomial characters, M-groups. The theorems of Taketa,  Dade and  Frobenius. Homogeneous components of completely reducible representations. Conjugate representations of normal subgroups. Inertia subgroups. Clifford’s theorem on the restriction of irreducible representations to normal subgroups. The theorem of Ito. Characterization of groups, all of whose character degrees are powers of  p.

Extending invariant characters of normal subgroups. If G/N is cyclic then all G-invariant characters of N can be extended to G. Extending linear characters. Nilpotence is determined by the group algebra. Hawkes’ construction of two groups with isomorphic group algebras such that exactly one of the two groups is supersolvable. Brauer’s characterization of characters. Quasi-elementary subgroups. Solomon’s theorem. Elementary subgroups. Applications: the values of a character of p-defect 0 on p-singular elements; The cyclotomic field corresponding to the exponent of the group is a splitting field for all irreducible (complex) representations.

Projectíve representations. Schur-multiplier. Extending representations to projective representations and characterizing those projective extensions equivalent to an ordinary representation.  Crossed group algebra. Finiteness of the Schur-multiplier. Central extensions. Schur’s theorem on the existence of the representation group. Irreducible representations of the symmetric group. Young tables. The character of a representation.

Getting the structure constants of the algebra of all class sums by means of the characters. Decomposition of the square of a character to symmetric and antisymmetric components. The number of involutions, bounding the size of conjugacy classes and character degrees. Applications: Groups of even order always have ``large’’ proper subgroups. There are only finitely many simple groups with given centralizer of an involution. The characterization of A5 by the centralizer of an involution.


Isaacs: Character Theory of Finite Groups

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