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 1 year 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...

Consider the group G = {1, −1, i, −i, j, −j, k, −k} under multiplication. Here i2= j2= k2= ijk = −1. determine which of the following sets is a subgroup of G. If a set is not a subgroup, give one reason why it is not. (a) {1, −1} (b) {i, −i, j, −j} (c) {1, −1, i, −i} (d) {1, i, −i, j}

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

Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G, prove that the normalizer of N(H) is the largest subgroup K of G such that HEK. In particular, if G is finite show that |H| divides |N(H)|. b) Show that H≤G⇐⇒every element B∈OH is a subgroup of G with B∼=H. c) Prove that HEG⇐⇒|OH|= 1.

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.

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

Let R be a relation on a set that is reflexive and symmetric but not transitive? Let R(x) = {y : x R y}. [Note that R(x) is the same as x / R except that R is not an equivalence relation in this case.] Does the set A = {R(x) : x ∈ A} always/sometimes/never form a partition of A? Prove that your answer is correct. Do not prove by examples.

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and...

Let L = {aibj | i ≠ j; i, j ≥ 0}. Design a CFG and a PDA for this language. Provide a direct design for both CFG and PDA (no conversions from one form to another allowed).

There exists a group G of order 8 having the following presentation: G=〈i,j,k | ij=k, jk=I,...

There exists a group G of order 8 having the following presentation: G=〈i,j,k | ij=k, jk=I, ki=j, i^2 =j^2 =k^2〉. Denotei2 bym. Showthat every element of G can be written in the form e, i, j, k, m, mi, mj, mk, and hence that these are precisely the distinct elements of G. Furthermore, write out the multiplication table for G (really, this should be going on while you do the first part of the problem).

def longest(string): start=0;end=1;i=0; while i<len(string): j=i+1 while j<len(string) and string[j]>string[j-1]: j+=1 if end-start<j-i: #update if current...

def longest(string): start=0;end=1;i=0; while i<len(string): j=i+1 while j<len(string) and string[j]>string[j-1]: j+=1 if end-start<j-i: #update if current string has greater length than #max start and end end=j start=i i=j; avg=0 for i in string[start:end]: avg+=int(i) print('The longest string in ascending order is',string[start:end]) print('Teh average is',avg/(end-start)) s=input('Enter a string') longest(s) i need a definition and explanation of this code that how it works?

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