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a. Show that any odd permutation is not the product of 3-cycles. b. Show that...

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a. Show that any odd permutation is not the product of 3-cycles. b. Show that there are either n or n/2 even permutations in any subgroup of order n of a symmetric group. Hint: consider a homomorphism to Z_2. What is the kernel? c. Using part b. or otherwise show that any element of odd order in a symmetric group is an even permutation. a. Show that any odd permutation is not the product of 3-cycles. b. Show that there are either n or n/2 even permutations in any subgroup of order n of a symmetric group. Hint: consider a homomorphism to Z_2. What is the kernel? c. Using part b. or otherwise show that any element of odd order in a symmetric group is an even permutation

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