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Problem 3. Let G be a group. a) Let [IOTA] e(x) = x-1 for all x in G. i. Show that , € Sym(G). ii. Suppose G has order n and let a = ca(2). What is the cycle type of c? iii. Deduce that if G has even order then G has an element of order 2. b) Fix g E G and define [RHO] p(x) = g.r for all x in G. i. Show that P E Sym(G). ii. Suppose G has order n and let b = o(g). What is the cycle type of p? iii. Deduce that o(g) I o(G) without appealing to Lagrange's Theorem. Problem 4. Fill in the following table for S6 like we did for Sa in Lecture 11. partition cycle type order number : : : : Using the table, determine (d) for all d. What is exp(S6)? Problem 5. Find explicit elements of S = Sym(N) with the following cycle types: a) (1,1,1, ;0) b) 1) c) (1,1,1, 1) d) (bonus) (0,0,0, 00). [Your answers must be unambiguous. That does not mean you need to come up with formulas or piecewise definitions; you just need to make it clear what the output would be given any positive integer input. For example, you can write out a small part of each permutation (using two-line or cycle notation) and indicate the pattern or the rest of the definition in words. No proofs required. The bonus must be flawless to count as correct.]

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