a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈...

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈ {1, 2, 3, 4, 5}}.

S5 will also act on this set. Consider the subgroup H = <(1, 2)(3, 4), (1, 3)(2, 4)>≤ S5. (a) Find the orbits of H in this action. Justify your answers. (b)

For each orbit find the stabiliser one of its members. Justify your answers.

action is this t.(i,j)=(t(i),t(j))

Expert Solution

Colby Messinger answered 2 years ago

Consider the group homomorphism φ : S3 × S5→ S5 and φ((σ, τ )) = τ...

Consider the group homomorphism φ : S3 × S5→ S5 and φ((σ, τ )) = τ . (a) Determine the kernel of φ. Prove your answer. Call K the kernel. (b) What are all the left cosets of K in S3× S5 using set builder notation. (c) What are all the right cosets of K in S3 × S5 using set builder notation. (d) What is the preimage of an element σ ∈ S5 under φ? (e) Compare your answers...

3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) = (σ(i), σ(j))....

3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) = (σ(i), σ(j)). (a) Describe the orbits of this action. (b) Show this is a faithful action, i.e. that the permutation represen- tation φ:S3 →SA =S9 (c) For each σ ∈ S3, find the cycle decomposition of φ(σ) in S9.

A) Suppose a group G has order 35 and acts on a set S consisting of...

A) Suppose a group G has order 35 and acts on a set S consisting of four elements. What can you say about the action? B) What happens if |G|=28? |G|=30?

Consider the group G = {1, −1, i, −i, j, −j, k, −k} under multiplication. Here...

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Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G,...

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We consider the operation of the symmetric group S4 on the set R[x,y,z,a] through permutation of...

We consider the operation of the symmetric group S4 on the set R[x,y,z,a] through permutation of an unknown integer. a) Calculate the length of the orbit of polynomial x2+y2+z+a. How many permutations leave this polynomial unchanged? b) Is a polynomial of length 5 under this operation possible? c) Show the existence of polynomials with orbit length 12 and 4.

i j(i) i j(i) i j(i) i j(i) 0 -0.0499 7 -0.08539 13 0.144812 19 0.08021...

i j(i) i j(i) i j(i) i j(i) 0 -0.0499 7 -0.08539 13 0.144812 19 0.08021 1 0.107506 8 0.062922 14 -0.0499 20 -0.30103 2 -0.06719 9 -0.04444 15 -0.18366 21 -0.33834 3 -0.04717 10 0.219422 16 -0.02898 22 0.058373 4 -0.09176 11 0.083849 17 0.08021 23 0.79083 5 -0.25918 12 -0.02261 18 -0.14271 24 0.130254 6 0.055643 25 -0.10177 (3 points) Please encipher the following plaintext with Caesar Cipher and the encryption key of 10: TODAYISTUESDAY; (3 points) In...

for (i=0; i<n; i++) for (j=1; j<n; j=j*2) for (k=0; k<j; k++) ... // Constant...

for (i=0; i<n; i++) for (j=1; j<n; j=j*2) for (k=0; k<j; k++) ... // Constant time operations end for end for end for Analyze the following code and provide a "Big-O" estimate of its running time in terms of n. Explain your analysis. Note: Credit will not be given only for answers - show all your work: (2 points) steps you took to get your answer. (1 point) your answer.

Let R be a relation on a set that is reflexive and symmetric but not transitive?...

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