1. (a) Compute the conjugacy classes of Qs. For each conjugacy class, compute the centralizer of an element of that conjugacy class and its index. Verify that the class equation holds. (b) Compute the conjugacy classes of DA. For each conjugacy class, compute the centralizer of an element of that conjugacy class and its index. Verify that the class equation holds. (c) Let G = GL, (R). Show that there is a group action of G on R" which acts by matrix multiplication. In other words, a(g, v) = 9(7) is a group action. (d) Let G = Sn. We define a map a : Sn X R" -> R", (0, (21, . .., 2n) ) > (25-1(1),...,20-1(n)) in other words, o permutes the coordinates of points. Show that this is a group action . (e) Let Be be the set of all subsets of {1, . .., n} of size k _ n. Show that there is a group action of Sn on Be, and therefore there is a homorphism Sn - S(my for every k En. Describe this homomorphism the case where n = 3 and k = 2