1.  [6 marks]. Consider on the set ℤ × ℤ the operation ∗ defined by

( + , ℓ − )  if is odd.

Show that (ℤ × ℤ, ∗) is a group. You may assume that ∗ is an associative operation.

2.Let be a group, and let ∶ → be the function with () = −1.

(a)  [3 marks]. Show that if is abelian, then the function is a group isomorphism. (b)  [2 marks]. Show that, conversely, if is a group isomorphism then is abelian.

3. Consider the elements , ∈ 5 given in two-line notation by

(a)  [2 marks]. Write and in cycle notation, and compute their product. (Remember that for us, ∗ means“first do , then ”)

(b)  [1 marks]. Compute the conjugate permutation = .

(c)  [2 marks]. Find another permutation , different from , such that = .

(d)∗ [1 mark]. In total, how many permutations are there which, like and , conjugate into ?

4.(a)  [2 marks]. Explain why a permutation is even if and only if it contains an even number of even- length cycles.

(b)  [2 marks]. List all the even cycle-types (apart from the identity) in 6.

(c)  [4 marks]. Calculate the number of elements of each of these cycle-types. Show your working, and briefly explain how you get your answers. (Make sure that, together with the identity, they sum to the order of 6, which is 360.)

(d)∗ [2 marks]. Give two different explanations of the following fact:

For any permutations , ∈ , the parity of is the same as the parity of the conjugate .

5.Let , ∈ 2(ℂ) be the complex-valued 2 × 2 matrices given by:

= (     ) = (    )

where is the complex number /4.

(a)  [3 marks]. Show that has order 2 and has order 8 in 2(ℂ).

(b)  [2 marks]. Verify the equation = 3 .