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MATH42061/62061 Coursework 1 DEADLINE: Thursday, 10th November by 4pm in reception. 1) Write down the character table for the Klein four-group, V4 = C2 × C2 . 2) The quaternion group of order 8 is the group you are familiar with on {±1, ±i ± j, ±k}, or Q8 = gphc, d | c4 = 1, d2 = c2 , dcd−1 = c−1 i. a) Show that c2 (or −1) generates a normal subgroup of Q8 and the quotient is isomorphic to V4 b) Calculate the conjugacy classes of Q8 . c) Use part (a) to start the writing down the character table of Q8 , then complete it using the orthogonality relations. 3) a) Show that if V is a G-space with a corresponding matrix representation ρV , then g 7→ [det(ρV (g))] gives a 1-dimensional matrix representation of G. b) Prove that this 1-dimensional representation does not depend on the basis chosen for ρV . Thus we have a well-defined 1-dimensional G-space; we denote it by det V and its character by χdet V or det χV . c) Find an expression for det χV (g) in terms of the eigenvalues of ρV (g). d) Calculate det χ for the 2-dimensional character from (2), identifying the row in the character table to which it corresponds (you can’t entirely avoid matrices). e) Suppose that for some χ and some g we have det χ(g) = −1. Show that G has a normal subgroup of index 2. 4) The centre of a group G is Z(G) = {h ∈ G | hg = gh for all g ∈ G}. You may quote any result from the course without proof. a) Let V be an irreducible G-space with matrix representation ρV . Show that if c ∈ Z(G) then ρV (c) is multiplication by a root of unity, λc , say. b)Deduce that χV (cg) = λc χV (g) for any g ∈ G. c) If V is also faithful, deduce that Z(G) must be cyclic. 1