6. Show that a cycle of odd length is an even permutation, and a cycle of even length is an odd permutation. Hence determine which of the following members of S10 are even, or odd: (i) ( 193 ) ( 26 ) ( 4 5 10 ) (78) (ii) ( 19326 ) ( 4 5 10 78), (iii) ( 1932645 10).

4. (i) Write the following permutations in S9 as a product of disjoint cycles 1 2 3 4 56789 f 7 298 34 165 ) , g= ( 1 2 3 4 5 6 7 8 9 56789 1 2 3 4 (ii) Calculate fg and write it both in two row notation and as a product of disjoint cycles. (iii) Calculate f-1 and write it both in two row notation and as a product of disjoint cycles. (iv) Find the order of f, and compute f20 as a product of disjoint cycles. (v) Determine which of f, g and fg are even or odd.

The permutations p = (3 5 4), q = (1 5 2 3) and r = (1 5 2) are elements of S5. (a) Find each of the following as a permutation in cycle form: wold wode ei doang poq, poqop-¹, q². (b) State the order and parity of each of po q and q². (c) Determine, in cycle form, two permutations in S5 that conjugate p to r.

Suppose a permutation s swaps the first two elements of {1, 2, ..., n}. A permutation r rotates the elements: r(1)=2, r(2)=3, ... r(n-1)=n, r(n)=1. We can thus write in cycle notation: s=(1 2), r=(1 2 3 4 ... n). What does the permutation rm sr -m do?

4. (i) Write the following permutations in S9 as a product of disjoint cycles 6 1 2 3 4 5 6 7 8 9 7 29 8 3 4 1 6 5 1 2 3 4 5 6 7 8 5 9 1 f = , 9 = 7 8 9 2 3 4 (ii) Calculate fg and write it both in two row notation and as a product of disjoint cycles. (iii) Calculate f-¹ and write it both in two row notation and as a product of disjoint cycles. (iv) Find the order of f, and compute f20 as a product of disjoint cycles. (v) Determine which of f, g and fg are even or odd.

(i) Write the following permutations in S9 as a product of disjoint cycles 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 789 7 298 3 4 1 65 5 6 7 89 1 234 f = 2 g= = (ii) Calculate fg and write it both in two row notation and as a product of disjoint cycles. (iii) Calculate f-¹ and write it both in two row notation and as a product of disjoint cycles. (iv) Find the order of ƒ, and compute ƒ2⁰ as a product of disjoint cycles. (v) Determine which of f, g and fg are even or odd.

Please do Exercise 14.4.21 part B and D. Please show step by step

An organization wants to assemble a cluster of n computers using wires and hubs. Each computer must have exactly one wire attached to it, while each hub can have up to three wires attached. There must be a path of wires between every pair of computers in the cluster. Based on this information, [(n – 2)/3] hubs are sufficient for the organization to assemble the cluster of n computers. True False

Compute σ0(1000), σ0(750), and σ0(1024)

5. 1.5, 2.25, 3, 3.75, ... Recursive: 16 Explicit: