Section A

Q 1. Consider the permutations f = (1 2 4)(5 7 8)(3 6), g = (1 2)(3 6)(4 5)(7 8) and h = (1 5)(2 7)(4 8) in S8.

(a) Write f, g and h in table form, all in one table.

(b) Calculate fg and gfg. Give your answers in cycle notation.

(c) Let S be your student number. Calculate fS and write it in cycle notation.

(d) Write down a permutation in S8 that has order 15.

(e) Does S8 contain an element of order 16? Explain your answer.

(f) Consider the set of all products you can make from f and g (such as fg, gfg, ggfffgffg, etc.) This is called the subgroup H of S8 generated by f and g. Determine whether H∼= Z6 × Z2, giving reasons for your answer.