Download MATH42061/62061 Coursework 1

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.
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