Question 5 (Units 132 and 133) – marks 1: se the subgroup test (Theorem 1324. Handbook page 38) to determine whether each of the following is a. subgroup of the symmetric group S4. (a) 7’ho. Hi of all permutations of {1.2,3.4} that fix the symbols 2 and a.
(61 The set H2 of all permutations of {1.2,3.4} that interchange the symbols 2 and 4.
t71
Question 6 (Unit 133) – 19 marks The two-line symbols for permutations p and q in S6 are
(1 2 3 4 5 6) P _3 4 6 2 5 1)
(1 2 3 4 5 and a = 2 5 3 4 6 1)
(a) Write down the permutations p, q, 20-1, q-1, p o q and q o p in cycle form. (b) Write down the orders of the permutations p, q, p o q and q o p. (c) Write down p and q as composites of transpositions, and state the parity of p and of q. (d) Determine, in cycle form, the elements of the cyclic subgroup H of S6 generated by p. (e) Find all the permutations in S6 that conjugate p to itself; that is, all elements g in S6 such that gopog-1=p.
Question 7 (Unit B4) — 8 marks Let x and y be elements of a finite group C, where x has order 3 and y has order 2. (a) Explain what can be deduced about the order of the group G. (b) Show that if yx = x2y then the elements xy and x2y are both self-inverse.
Question 8 (Book B) – 5 marks Five marks on this assignment are allocated for good mathematical communication in your answers to Questions 1 to 7. You do not have to submit any extra work for Question 8, but you should check through your assignment carefully. making sure that you have explained your reasoning clearly, used notation correctly and written in proper sentences.