Here are the problem: Consider the following odd permutations f in the permutati

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Here are the problem:

Consider the following odd permutations f in the permutation group S7: f: 1 3, 2 6, 3 0, 4 7, 5 , 6 2, 7 1. Determine f by writing f as a product of disjoint cycles. Find o(f). Compute f32, express f32 as a product of disjoint cycles. Hint. The fact that f S7 should narrow it down to 2 possibilities, out of which only one yields an odd permutation. Once you know f, the rest is straightforward. Find permutations as required, or otherwise state that they do not exist. Find a permutation f S15 such that o(f) = 56. Find an even permutation g S15 such that o(g) = 36. Find an even permutation h S13 such that o(h) = 30. Hint. Write each of f, g and h as a product of disjoint cycles. For example, if we let phi = (1 2 3 4 5)(6 7) Sl0 and psi = (1 2 3 4 5)(6 7)(8 9) S10, then o(phi) = lcm(5,2) = 10, o(psi) = lcm(5, 2,2) = 10, phi is odd, and psi is even.

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