2. What are the irreducible representations of the same (cyclic, order-N) group G over Q? Over a finite field of characteristic not dividing N?
3. Fix a prime power q, and let and G be the ax+b group over a finite field F of q elements. Show that G has q-1 one-dimensional representations over C (use the homomorphism b from G to the multiplicative group of F). Show also that G has q conjugacy classes. Therefore G has just one more representation, whose dimension must be q-1 by the sum-of-squares formula. Find such a representation explicitly (Hint: start from the permutation representation of G on Cq coming from the action of G on F). Determine the character of this extra representation, and verify that its inner product with itself equals 1.
4. Determine the simple factors and irreducible representations of k[G] where G is either of the two non-commutative groups of 8 elements (viz., the dihedral and quaternion groups), and k is either C or R.