2 MATH 150C, 2016 SPRING DUE DATE: (MONDAY) 4/11/2016 Suggested readings: Section 10.2, Section 10.3, Section 10.6 (before Lemma 10.6.3), Section 10.7 (before Theorem 10.7.6) Do: (1) Problem 10.2.3 (2) Problem 10.3.5 (3) Let G = {g1 , , gn } be a nite group. Consider the following CG module V, where n i each element v V has the form v = i gi , where i C . In other words, the gi form a basis for V . Addition and scalar multiplication are dened in the obvious way. The group G acts on the vector space in the obvious way: g(v) = n i i g(gi ) for all g G. With respect to the basis {gi }, we have a representation (g) = [g]G , where [g]G is the matrix representation with respect to the basis G. Show that this is a faithful representation. This is called the regular representation of G. Compare with the beginning of Section 10.6. (4) Let G = Z2 Z2 (Recall that this is an abelian group with 4 elements and that it is not cyclic. You don't need this for this exercise). (a) Write down the regular representations for G. (d) Let v = g1 +g2 +g3 +g4 be an element in the CG-module. Show that v2 = 4v. (5) Use Problem (3) to show that we can always nd a faithful, irreducible CG-module if G is a nite simple group. (Recall that a group is simple if {e} and G are its only normal subgroups.) (6) In class, we showed that given an FG-module V and a projection : V V, we have V = Ker Im. Show (by an example) that this is not true if is just an FG-homomorphism. Date: April 3, 2016. 1