Question

2. Consider the permutations a (123)(45) and b (2543) in the symmetric group S (a) Compute the conjugate permutation ca using (i) the definition a-b ab (b) What is the order of a? How many permutations have the same shape as a; that is, (x x x)(x x). (c) What is the subgroup H of all permutations in Ss that commute with the permutation a? d) Using the result of the previous part, or otherwise, find 5 other permutations bi, b2, , bz, different (e) Could there be more than 6 different p andi) the faster method shown in lectures Note how this number is a factor of S As well as using symbols, deseribe this subgroup briefly in terms of the cycle structure of a from b, such that a-c. This should give 6 permutations, all of which conjugate a intoc. Give a brief explanation of your answer, perhaps referring to the subgroup H of part (c) and the conjugating a into c as in the previous part? ermutations number in part (b).

Answer 1

43) eSS (3452)い23705) (25y 3) 2 S2 34 12 3 4 ao b 오 3159 ) ( 234 よ43.15 (124) (3)(11 4)

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