Representation Theory

1. Throughout this question, G denotes the group with the following presentation: (x,y,z|24 = 1, x2 = z, y2 = 23, zx = xz, zy = yz, yx = ryz?). You can take for granted that G has 16 elements, divided into 10 conjugacy classes as follows: {1}, {z}, {z?}, {z}, {r, xz}, {r2, 22}, {y, yz?}, {yz,yz}, {ry, tyz?}, {ryz, ryz3}. (a) Let S be a simple CG-module. Explain why z must act on S as multiplication by one of the following complex numbers: 1, -1, i, -i. (b) Show that there is some simple CG-module on which z acts as multiplication by i. (c) Determine the character table of G, explaining your reasoning. (d) Let V be a simple CG-module on which z acts as multiplication by i, and let p: G + GL(V) be the corresponding group homomorphism. Find the eigenvalues of p(x) and py). 1. Throughout this question, G denotes the group with the following presentation: (x,y,z|24 = 1, x2 = z, y2 = 23, zx = xz, zy = yz, yx = ryz?). You can take for granted that G has 16 elements, divided into 10 conjugacy classes as follows: {1}, {z}, {z?}, {z}, {r, xz}, {r2, 22}, {y, yz?}, {yz,yz}, {ry, tyz?}, {ryz, ryz3}. (a) Let S be a simple CG-module. Explain why z must act on S as multiplication by one of the following complex numbers: 1, -1, i, -i. (b) Show that there is some simple CG-module on which z acts as multiplication by i. (c) Determine the character table of G, explaining your reasoning. (d) Let V be a simple CG-module on which z acts as multiplication by i, and let p: G + GL(V) be the corresponding group homomorphism. Find the eigenvalues of p(x) and py)