Representation & Finite Groups

MA4142 REPRESENTATION THEORY OF FINITE GROUPS PROBLEM SHEET 2 This is the second problems sheet. The deadline for submitting so- lutions is 3th December, 10.00. Total marks of correct solutions is 30 marks. 1) i) Let G be a nite group and g 2 G. Prove that if g is central (that is xg = gx for all x 2 G), then j(g)j = j(1)j for every irreducible character . ii) Show that if G has a irreducible representation : G ! GL(V ) such that Ker() = 1, then the center of G is a cyclic group. 2) The character table of D4 is given by 1 1 2 2 2 G 1 x2 x y xy 1 1 1 1 1 1 2 1 1 1 ??1 ??1 3 1 1 ??1 1 ??1 4 1 1 ??1 ??1 1 5 2 ??2 0 0 0 i) Let f and g be a class functions given by f(1) = f(x) = 1; f(x2) = 3; f(y) = f(xy) = 0; g(1) = 4; g(x2) = 10; g(x) = g(y) = g(xy) = 0: Is either of these functions the character of a representation? If yes, then nd the corresponding representation. ii) Compute the ring R(D4) and the Adams operations therein. 3) Let G be the group of order 16 dened in terms of generators and relations G =< x; y : x4 = 1 = y4; yx = x3y > Find character table of G. Compute the ring R(G). 1 2 PROBLEM SHEET 2