Group Representation Theory

(1) Let G:= <g|g^4 = 1> and F be the field of real numbers. Show that the representation

p:g -> (0 -1)

(-1 0)

is irreducible over F.

(2) Let G be a finite group, x & y two characters of G

(i) If x is linear, show the x.y is irreducible <=> y is irreducible

(ii) If y(1) > 1, show that y .y is always irreducible.

(iii)Let x be an element of Irreducible G and no other character in Irreducible G have the same degree as x. Suppose that there is a linear character y such that for some g, y(G) is not equal to 1. Show that x(g) = 0.