View S3 as a subset of S5 in the obvious way. For σ, τ ∈ S5,...

View S₃ as a subset of S₅ in the obvious way. For σ, τ ∈ S₅, define σ ∼ τ if στ ^-1 ∈ S₃.

(a) Prove that ∼ is an equivalence relation on S5.

(b) Find the equivalence class of (4, 5).

(c) Find the equivalence class of (1, 2, 3, 4, 5).

(d) Determine the total number of equivalence classes

Expert Solution

Colby Messinger answered 3 years ago

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

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Convert in machine code for MIPS LW S5, 33(S2); Add R1, T3, S5; Sub T4, S3,...

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( X, τ ) is normal if and only if for each closed subset C of...

( X, τ ) is normal if and only if for each closed subset C of X and each open set U such that C ⊆ U, there exists an open set V satisfies C ⊆ V ⊆clu( V) ⊆ U

Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an...

Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an experiment having the probability distribution shown in the accompanying table. If A = {s1, s2} and B = {s1, s5, s6}, find the following. Outcome Probability s1 1 3 s2 1 7 s3 1 6 s4 1 6 s5 1 21 s6 1 7 (a) P(A) = P(B) = (b) P(AC) = P(BC) = (c) P(A ∩ B) = (d) P(A ∪ B) =...

(The “conjugation rewrite lemma”.) Let σ and τ be permutations. (a) Show that if σ maps...

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Consider the following spot rate curve: s1 s2 s3 s4 s5 0.050 0.055 0.061 0.066 0.075...

Consider the following spot rate curve: s1 s2 s3 s4 s5 0.050 0.055 0.061 0.066 0.075 (a) What is the forward interest rate that applies from period 3 to period 5? That is, what is the value of f3,5? Assume annual compounding. (Keep your answer to 4 decimal places, e.g. 0.1234.) (b) If the market forward rate from period 3 to period 5 is not equal to the value derived in (a), how can you create an arbitrage opportunity? (No...

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

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Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum...

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes). (a) Evaluate τ (1500) and σ(8!). (b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503. (c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).

Show that any two permutations σ,τ ∈ Sn have the same cycle structure if and only...

Show that any two permutations σ,τ ∈ Sn have the same cycle structure if and only if there exists γ ∈ Sn such that σ = γτγ^−1.

Write down every permutation in S3 as a product of 2-cycles in the most efficient way...

Write down every permutation in S3 as a product of 2-cycles in the most efficient way you can find (i.e., use the fewest possible transpositions). Now, write every permutation in S3 as a product of adjacent 2-cycles, but don’t worry about whether your decomposition are efficient. Any observations about the number of transpositions you used in each case? Think about even versus odd.

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