A few more examples/exercises on representations of finite groups

1. Let G be a cyclic group of order N, with generator g. Fix a primitive n-th root of unity w in C (the standard choice is e2 Pi i/N). For each m in Z/NZ, we then get a 1-dimensional (and thus necessarily irreducible) representation of G by mapping gn to wmn. These are all the irreducible representations of G (why?). This gives the decomposition of the group algebra C[G] as a direct sum of simple C-algebras, which in this case are all isomorphic to C. Interpret this, and the orthogonality theorems for characters of finite groups, in terms of the ``discrete Fourier transform'' on complex-valued functions on G.

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.