(3) Let 0 = 1234 6789 365 987 in S9 (a) Compute o2 and o (b) Write o as a product of disjoint cycles (c) Compute |o, the order of o (d) Write o as a product of transpositions (e) Is o even or odd?
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Please do part d) and e)
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- One advantage of decomposing a permutation as a product of disjoint cycles is that disjoint cycles commute. To verify this, let a = {a1, a2, ...., ak} and b = {b1,b2, ...., bm} be disjoint cycles in Sn. Show that (ab)(t) = (ba)(t) for every t ∈ {1,2,..., n} by considering cases.Find an exponential generating function for the number of permutations with repetition of length n of the set {a, b, c} in which there are an odd number of a’s, an even number of b’s, and an even number of c’s.Prove that Zn has an even number of generators if n > 2. What doesthis tell you about Φ(n)?
- Prove: For n > 1, n − 1 is a generator of Zn.Let (a1a2a3a4) and (a5a6) be disjoint cycles in S10. Show that there isno element x in S10 such that x2 = (a1a2a3a4)(a5a6).Compute the index of 〈3〉 in ℤ12. Then write down all of the cosets of 〈3〉 in ℤ12. Finally, write a Cayley table for ℤ12/〈3〉. Is this quotient an abelian?
- How many elements of order 4 does Z4 ⊕ Z4 have? (Do not do this by examining each element.) Explain why Z4 ⊕ Z4 has the same number of elements of order 4 as does Z8000000 ⊕ Z400000. Generalize to the case Zm ⊕ Zn.show that the set Z ={0,+-1, +-2,...} with the operation of multiplication (a,b) map to ab is not a group Show that the operation (a,b) map to a*b =ab on the set of positive real numbers is not associative. for the permutation sigma belongs to S8 sigma=(3 7 4 2 6 8 1 5). Find the number of inversions I(sigma) sgn sigma =(-1)sigma , the decomposition into a product of independent cycles and the order for the permutation sigma belongs to S8 tau=(7 4 8 1 3 5 6 2). Find the number of inversions I(sigma) sgn sigma =(-1)sigma , the decomposition into a product of independent cycles and the order find the permutation U belongs to S8 such that sigma *U = tau where sigma=(3 7 4 2 6 8 1 5), tau=(7 4 8 1 3 5 6 2) are the permutation. decompose U into product of indepent cycles, find the sign and order consider the group of permutations Sn. Let A={a1,..., ak} subset of {1; : : : ; n}be a set of k less than equal to n distinct elements. Let SA subset of Sn be the set of permutationspreserving the…Find number of elements of order 2 in A5 (Alternating group)