18.704 Problem Set 1
Due Tue 9/22,  at lecture.

We denote a two by two matrix with entries 

ab
cd

as (a&b//c&d).

1. Show that the elements 

1=(1&0//0&1), -1=(-1&0//0&-1),I=(i&0//0&-i),-I=(-i&0//0&i),
J=(0&1//-1&0),-J=(0&-1//1&0),K=(0&i//i&0),-K=(0&-i//-i&0)

form a subgroup Q of the group of two by two invertible matrices with complex
entries.

Show that IJ=K,JI=-K, JK=I, KJ=-I, KI=J, IK=-J, I^2=-1, J^2=-1, K^2=-1.

2. Find all homomorphisms from Q to C-\{0\}.

3. Let G be a finite group and let H,K be subgroups of G. 
For any x in G, let HxK be the set of all elements of G of the form hxk 
for some h in H, k in K. 
A subset of G of the form HxK is called a double coset. 
Does the order of a double coset always divide the size of the group G? 
Either prove this is true, or find a counterexample.