Exercises 1. In S,, show that we can find elements, a, b s.t., (ab) # a b2. Also show that there exist four elements satisfying x = e and three elements satisfying ys = e. 2. Find order of all elements in S, and list all its subgroups. 3. Show that S, has no element of order greater than 4. 4. Using Index theorem show that if a group G of order 35 has a subgroup H of order 7 then H is normal in G. 5. Let H = {I, (123), (132)}, K = {I, (12); be two subgroups of S,. Write all the left and right cosets of H and K in S,. Hence show that H is normal whereas K is not normal in S3. 6. Verifty Cayley's theorem for (i) a cyclic group of order 3. (ii) G = {1, -1, i, -i). 7. In S,, show that we can find elements a, b s.t., o(a) = o(b) = 3 and o(ab) = 5. [Take a = (123), b = (145)] 8. Let o = (1234) be a 4 cycle. Show that of is also a 4-cycle iff k and 4 are co-prime. Generalise! 9. Show that product of two transpositions is either (i) the identity or (ii) a 3-cycle or (iii) a product of two 3-cycles. (See problem 51). 10. Show that Ag contains an element of order 15. [Note (123), (45678) are even permutations]. 11. Show that KA 54 = {KA, K,(12), KA(13), KA(23), KA(123), KA(132) } [Hint: K,(14) = K,(23) as (14)(23) E KA, KA(124) = KA(132) as (124)(123) = (14)(23) E K, etc. ]