Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about the order of the element αβ if α and β are disjoint cycles? If the cycles α and β share a single element in common? If the cycles α and β share two elements in common? (There are two different answers to this last question!)

Q: QUESTION 23 There are 150 seats in the auditorium. 40 located on the left; 50 located on the right;…

A: At least 23 means 23 or 24 or 25.There are 60 seats which are located in the center. So (150-60)=90…

Q: Suppose that there are 6 people in a room. Show that one can always find a group of 3 people such…

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Q: Why is it that if I do row expansion on the first row, I get det(A) = -1, but, if I do it in the…

A: We need to find the determinant of a using the first row expansion and as well using the second…

Q: If (ab) and (cd) are distinct 2-cycles in Sn, prove that (ab) and (cd)commute if and only if they…

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Q: Show that if n>=m then the number of m-cycles in Sn is given by: n(n-1)(n-2)...(n-m+1)/m

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Q: Find the closed form solution of recurrence relation defined by T(1) = 2,T(n) = 2n · T(n – 1), n > 1…

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Q: Suppose that an < bn < Cn for all n, an → L, and cn → L. Show that bn → L.

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Q: Given any whole number w, how many whole number triple decompositions does w have? Prove it.

A: A number is said to be whole when it begins from 0 and has no end. Decomposition of number means…

Q: • If the chain is in state 1 on a given observation, then it is five times as likely to be in state…

A: From the given information,  If the chain is in state 1 on a given observation, then it is five…

Q: In (Z24, €) find the orders of the elements 3, 4, 6, 7, and 18.

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: Find the order of the residue class 5 in Z/29Z.

A:

Q: One day you saw a news on Facebook about COVID-19. At 1 PM. you tagged the post to 1% of your…

A: This problem can be considered as recurrence relation problem.

Q: This exercise provides an alternative to Theorem 8.2 for factoring an r-cycle into transpositions.…

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Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: Illustrate the Abel's Formula?

A: To  illustrate the Abel's Formula.

Q: Let R be a P.I.D. Show that every ascending chain of ideals (a1) < (a2) < (a;) ... C (a,) .. is…

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Q: Express the permutation (2 4 5)(1 3 5 5)(1 25) as a single cycle or as a product of cycles.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Suppose that b₁ = 2bn-1 +15bn-2 for n ≥ 2, with b = 5 and b₁ = 1. Solve this recursion to obtain a…

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Q: SHOW THE FULL SOLUTION OF THIS PERMUTATION 3!

A: Given3!

Q: Find the solution to the recurrence relation an=5an-1, n>=1 with initial condition ag= 1.

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Q: 5. If the pair of cycles a and ß have no entries in common, then aß= ßa. True False ●

A: answer is in next step

Q: In general, if (12 · · · k) is a cycle of length k and o is any permutation then o(12 ·.· k)o -1 =…

A: Given: T=(12.....k)

Q: a) What is the maximum possible order of an element in S,? Give an example of one of these elements.…

A: Given, Group S7 any element can be written in cycle notation i.e. in disjoint cycles as example…

Q: Suppose that there exists a neighborhood of k > 5 houses, arranged in a circle. The landlords want…

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Q: 1 2 2 3 4 5 6 7 8 2 1 6 3 5 7 8 4 2 1

A:

Q: Find the orders of all the elements of Ag.

A: A8 is the group of all even permutation of S8.

Q: Each year, 50% of the people from City A move to City B, and 30% of the people from City B move to…

A: We have to find the population of two cities after 2 years.

Q: Prove that every element of Sn (n &gt; 1) can be written as a productof elements of the form (1k).

A:

Q: A) Find at least 12 fractions n /403 that have exactly 6 digits in their repetends . In their…

A: Since, you have post multiple question. I am answering first one. Please post again for the…

Q: An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of every…

A: Given a n×n magic square

Q: QUESTION 13 List all the elements of order 12 in Z600.

A:

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: The product of an even number of disjoint cyclic is an even permutation

A: True

Q: If o is a cycle of length n, then a' is also a cycle if and only if... 50) Show that S is generated…

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Q: Suppose we build an LFSR machine that works mod 3 instead of mod 2. It uses a recurrence of length 2…

A: Consider the given equation. xn+2≡c0xn+c1xn+1mod3 And, the given values are defined as under modules…

Q: Suppose that n > 2 is an even integer and a is an (n – 1) -cycle in S,. Show that a does not commute…

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Q: 4. Write the permutation (6 5 7 9 2 1 4 8 (1 2 3 4 5 6 7 8 9. 10 as a product of disjoint cycles 10…

A: A permutation is a kind of arrangement of set of objects in order. To obtain the disjoint cycles,…

Q: Question 2 (Cyc Given positive integers a1, a2, ...,ak with a1 + a2 +...+ ag = n, how many…

A: Given that there are a1,a2,...,ak positive integers. Also a1+a2+...+ak=n. Without loss of generality…

Q: Show that an= 2n -5n is also a solution to the recurrence relation an= 7an-1 − 10an-2. What would…

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Q: QUESTION 5 Find all the elements of order 12 in Z48000 -

A: Number of elements of [m] in Zn is phi(m)

Q: Find the all 7 principal minors (PMs) of (I- A)

A:

Q: 4. Show that if ð > 2, then G contains a cycle.

A: We need to prove that δ≥2 , then G contains a cycle , Asume that G is a finite graph, Let V1 ∈ G,…

Q: Question 5. - J (a) Express the permutation (2 4 5) (1 3 5 4) (1 2 5) as a single cycle or a s a…

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Q: Find the unique, closed form solution to this recurrence relation given the initial conditions: an =…

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Q: Find the simpler circuits with the same output of the following sum-of-products expansion. xyz +…

A: We will simplify the following boolean expression.

Q: Show that for A and B of n×n order with complex enteries the matric Ab need not be similar to BA ,…

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Q: Q/ Z is infinite, but every element of Q/ Z has a finite order.

A: ℚℤ is infinite, but every element of ℚℤhas a finite order.Now consider the group of rational number…

Q: Suppose n(X)=240n(X)=240 and XX is partitioned into six subsets X1X1, X2X2, X3X3, X4X4, X5X5, and…

A: The given details aren(X) = 240n(X1) = n(X2) = n(X3) = xn(X4) = n(X5) = n(X6) = y n(X1) = 3n(X5),…