1. Let D be 2. Let G = {a + b

D : a, b Q} \ {0}.

Define multiplication by (a + b

D)(c + d

D) = (ac bdD) + (ad + bc)

D. Show that

G is a group with respect to multiplication.

2. Let r be 5 Let H = {

x y

y x + ry

: x, y R}.

Show that H is a subgroup of GL2(R).

3. Let n be 2. Let G = {k N : 1 k n, gcd(k, n) = 1}.

Then G forms a group under multiplication modulo n. Write down the Cayley table for this

group, and determine whether or not the group is cyclic.

4. Let's do some permutation calculations.

(a) =(2051)(89)(92)

(b) =(205)(189)(92). Compute

2

1

, giving the answer as a

product of disjoint cycles.

5. n = 24. k = 3. Give an example of a

group G of order n and a subgroup H of order k. Then list all of the cosets of G/H.