answer all parts please

3. Consider the group U(20) = {n E Z20 | ged(n, 20) = 1}. Prove that U(20) is not cyclic. 4. Let GL(2, C) be the group of 2 x 2 complex invertible matrices. Write Let D = (a, B) be the smallest subgroup of GL(2, C) containing a, B. Let H = (a, y) be the smallest subgroup of GL(2, C) containing a, y. (a) Prove that H has exactly 8 elements, and in fact, H = {+I, +x, 1, y, tz} for some x, y, z E GL(2, C) which satisfy 12 = 12 = 22 = -1, xy = z, yz = 1, zx=y. (b) Consider the Cayley table for the group of quaternions Q: given on page 95 of Gallian. Is there a way to rename the elements of H so as to make the Cayley tables of H and Q8 identical? (c) Is the group D finite? How many elements are in D? (d) Is there a connection between D and D4? Prove any assertion you make. (e) Prove that there is no way to rename the elements of D to obtain the Cayley table of H. Hint: What are the orders of the elements of D and H, and how many elements are there of each order