The 'cycle structure' of a permutation of a given finite set X is an expression of...
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The 'cycle structure' of a permutation of a given finite set X is an expression of the form kk...km where m, is the number of k-cycles of the disjoint cycle representation for , for 1 ≤ i ≤s. Usually such terms with m, = 0 are ignored. Hence, for example, the permutation (1, 2,5) (4, 6, 7, 8, 9) (12, 13, 14) in S₁4 has cycle structure 1³ 3² 5¹. (a) (3 marks) What are the possible cycle structures for an element of the symmetric group S₂? Also give the number of elements of S, with that cycle structure in each case. (b) (2 marks) Which of the possibilities in part (a) can be written as a product of an even number of transpositions? (c) (4 marks) What are the possible cycle structures for an element of the symmetric group S₂? Also give the number of elements of S, with that cycle structure in each case. (d) (3 marks) What are the possible cycle structures for an element of the symmetric group Se having order 2? Also give the number of elements of S, with that cycle structure in each case. (e) (2 marks) What are the possible cycle structures for an element of the symmetric group S having no fixed point? The 'cycle structure' of a permutation of a given finite set X is an expression of the form kk...km where m, is the number of k-cycles of the disjoint cycle representation for , for 1 ≤ i ≤s. Usually such terms with m, = 0 are ignored. Hence, for example, the permutation (1, 2,5) (4, 6, 7, 8, 9) (12, 13, 14) in S₁4 has cycle structure 1³ 3² 5¹. (a) (3 marks) What are the possible cycle structures for an element of the symmetric group S₂? Also give the number of elements of S, with that cycle structure in each case. (b) (2 marks) Which of the possibilities in part (a) can be written as a product of an even number of transpositions? (c) (4 marks) What are the possible cycle structures for an element of the symmetric group S₂? Also give the number of elements of S, with that cycle structure in each case. (d) (3 marks) What are the possible cycle structures for an element of the symmetric group Se having order 2? Also give the number of elements of S, with that cycle structure in each case. (e) (2 marks) What are the possible cycle structures for an element of the symmetric group S having no fixed point?
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