Let D be 2. Let G = {a + bD : a, b Q} \ {0}.Define multiplication by (a + bD)(c + dD) = (ac bdD) + (ad + bc)D. Show that G is a group with respect to multiplication.

2. Let r be 5 Let H = {x yy 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.