use cycle notation

(2) (12 points) Working in the Group S8, in each part below, find a permutation that satisfies the required property. (i) Find a permutation ke Sg which is a product of 2 disjoint cycles and ord(K) = 10. Be sure to show that your order is exactly 10 and list the elements of the cyclic subgroup (K). (ii) Find a permutation we Sg which is the product of at least 2 disjoint cycles so that w is an odd permutation AND ord(w) is an even integer. Verify that w has the properties needed. (iii) Find a permutation de S8, which is the product of at least 2 disjoint cycles so that S is an EVEN permutation AND ord() is an odd integer. Verify that has the properties needed. (2) (12 points) Working in the Group S8, in each part below, find a permutation that satisfies the required property. (i) Find a permutation ke Sg which is a product of 2 disjoint cycles and ord(K) = 10. Be sure to show that your order is exactly 10 and list the elements of the cyclic subgroup (K). (ii) Find a permutation we Sg which is the product of at least 2 disjoint cycles so that w is an odd permutation AND ord(w) is an even integer. Verify that w has the properties needed. (iii) Find a permutation de S8, which is the product of at least 2 disjoint cycles so that S is an EVEN permutation AND ord() is an odd integer. Verify that has the properties needed