Homework 3. Due Feb.28

(1) Let (s_i) be a set of representatives for the conjugacy classes of G (it contains one
element in each conjugacy class) Let \chi:G@>>>C^* be an irreducible character of G.
Let n=\sum_i \chi(s_i). Show that n is an integer greater than or equal to 0. (Hint: show
that n is the inner product of \chi with the character of a permutation representation of G.)

(2) Let G be the group of permutations of {1,2,3,4}. Let A be the subset of G consisting of
the trivial permutation and the three permutations (12)(34),(13)(24),(14}(23). Here (ij)
denotes a trasposition interchanging i,j. Show that A is a normal subgroup of G.
Let H be the subgroup of G consisting of permutations which keep 4 fixed. Show that G is a
semidirect product of A and H. Using the known description of the irreducible representations
of a semidirect product, determine the list of degrees of the various irreducible representations
of G.