1. Define the mobius group, and describe how it acts on C U {infinity} , U is union.

Show that the subgroup of the mobius group consisting of transformations which fix 0 and infinity is isomorphic to C\{0}

Now show that the subgroup of the mobius group consisting of transformations which fix 0 and 1 is also isomorphic to C\{0}

2. let G be a dihedral group of order 12

i) list all the subgroups of G of order 2. Which of them are normal

ii) list all the remaining proper subgroups of G

iii) for each proper normal subgroup N of G , describe the quotient group G/N

iv) show that G is not isomorphic to the alternating group A4

3. Show that if a group G contains a normal subgroup of order 3, and a normal subgroup of order 5, then G contains an element of order 15.

Give an example of a group of order 10 with no element of order 10.

4. suppose that G is the group of rotational symmetries of a cube C. Two regular tetrahedra T and T' are inscribe in C, each using half the vertices of C. what is the order of the stabilizer in G of T?

5. Suppose that G is a finite group of rotations in R2 about the origin. Is G cyclic? justify your answer.

6. G X H = { ( g,h): g belongs to G, h belongs to H}

show how to make G X H into a group in such a way that G X H contains subgroups isomorphic to G and H.