Problem 4 Let o = (4573) and T = (46823) denote permutations in Sg (a.) Calculate the disjoint cycle decomposition for OT (b.) Find the order for OT (c.) Find all generators for (or) ≤ Sg. Express your answers in terms of (7)k for approrpriate k, no need to calculate the disjoint cycle decomposition for these generators.
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- Write out the addition and multiplication tables for 5.If o is a k-cycle then the order of o is k. On the otherhand, if 7 is a product of disjoint cycles o1, 02, 03 .m such that the order of o; is k; then the order of 7 is given by lem(k1, k2, kg, ..., km). Find the order each of the following. (a) (1 32 4) e S4. (b) (1 4 3)(2 5 6 7) E S7. (c) (1 43 6)(2 9)(5 7 8) E Sg. (d) (1 7)(2 6)(3 5 4) o (1 2 3)(4 5 6 7) e S,Let a = (1432)(257)(654). (a) Find a disjoint cycle decomposition of a, i.e. write a as a product of disjoint cycles. (b) Find a transposition decomposition of a, i.e. write a as a product of transpositions. (c) What is the order of a? (d) Is a an even or odd permutation?
- Consider the permutation (taqgblvxh)(n skw fue)(ycodjr)(pz mi). (a) Showing your working, and working entirely in disjoint cycle notation, give the disjoint cycle notations for T and for T-1, and then compute T*(x) and -1(c). (b) Justifying your answer, find two positive powers a and B such that n° = T = T (c) Explain how a monoalphabetic substitution cipher works. Illustrate your explana- tion by using the permutation 7 above to encrypt the plaintext It is not that secrets are never needed in security, but they are never desirable.2. Sketch the graph of the quadratic y=-2(x+3)2 +8 by using a series of transformations. Show all of the transformed tables. hot mod, binuot erfass W ST gnist your spo(a) Write (152) (341) (6352) ≤ S6 in array form. Fill in the blank row below. 1 2 3 4 5 6 (b) Express [ 1 2 3 4 5 6 7 8 9 1 4 2 97 8 5 3 6 as a product of disjoint cycles. (c) How many elements of order 10 does S7 have?
- ToT. (o and 8 are called conjugate elements) Let o,TE Sn. Define 6 = Show that if o (i) = j, then 6 = (t(i)) = T(j). (a) (b) Explain that the previous part says that if you apply T to each of the entries in the cycle notation of o, then you get the cycle notation for ô. In other words, if o has cycle decomposition (а, а .. ак, )(b bz ... bk,).... Then 8 has cycle decomposition (143 8) (2 6 5), (163) 7 5 2), Illustrate the previous part by letting (c) σ Ξ and quickly writing down the cycle decomposition for toT1. Check yourhow to get the wronskian of the following?(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.
- 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?1. Fifteen balls, including three each of five different colors, are arrange in a triangle as shown. How many ways can this be done if arrangements obtained by rotations are considered the same? 2. (a) Construct the cycle index polynomial for coloring the faces of the regular tetrahedron. (There are 12 rotations including the identity that map the regular tetrahedron, as a rigid body in 3-space, into itself.) (b) Find the number of ways to paint the faces of a tetrahedron using k colors, for each k = 2, 3, 4. 3. (a) Compute the cycle index polynomial for the group of rotations of the cube represented as permu- tations of the six faces. (There are 24 rotations including the identity that map the cube, as a rigid body in 3-space, into itself. We are ignoring the 24 reflections of the cube.) What is the number of essentially different ways to paint the faces of the cube in 2 colors? In 3 colors? In n colors? (b) What is the number of essentially different ways to paint the faces of a cube…(3) Let G = ( 6789 3 6 5 1 4 9 8 7 2 1 2 3 4 5 (a) Compute G²³ and or € Sq 9. (b) Worte o cycles (c) Compente 101 = ordrol the order do as a product of deyout ฝา di White o as a product of trampas. lel Is o or old. even