Mathman wrote:

1. U = {z is an element of C such that the absolute value of z = 1)

Show that U is a subgroup of (C, *).

To prove that a group H is a subgroup of another group G under multiplication, you need to show the following:

1) H is closed under muliplication

2) The identity element e of G is contained in H

3) If

is an element of H, then so is a^{-1}

Mathman wrote:

...To what groups is Un isomorphic?

My suggestion would be to multiply arbitrary elements of U_n together and see what you get. Start with small examples, such as U_4.