discipline 
Pure
Mathematics

subject 
Representation theory of groups

lecturers 
Péter
Hermann, Péter
Pál Pálfy, József Pelikán 
credits 
2 
period 

curriculum 
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: C_{G}(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, Mgroups. 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 Ginvariant 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. Quasielementary subgroups. Solomon’s theorem. Elementary subgroups. Applications: the values of a character of pdefect 0 on psingular elements; The cyclotomic field corresponding to the exponent of the group is a splitting field for all irreducible (complex) representations. Projectíve representations. Schurmultiplier. Extending representations to projective representations and characterizing those projective extensions equivalent to an ordinary representation. Crossed group algebra. Finiteness of the Schurmultiplier. 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 A_{5} by the centralizer of an involution. 
literature 
Isaacs:
Character Theory of Finite Groups 
form of tuition 
Lecture 
mode of assessment 
written/oral
exam 