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

Q: A student claims that he does not need to define permuta- tions because he knows the Fundamental…

A: A permutation is the arrangement of objects where the order of arrangement is considered. The…

Q: 14. Let A = {1,2,3,4,5,6} and p₁ = (4,2,5) and p₂ = (2,6,1) be permutations of A. Compute p₂¹P₁-1…

A:

Q: List all the elements of S4, written in cyclic notation.

A: To list out all the elements of S4.

Q: The modular representation of the integer 23 with respect to the modulo 2, 3, 5 is (1,2,3) (3,2,1)…

A: the modular representation of the integer 23 with respect to the modulo 2,3,5 is it is known that…

Q: What elements are generating in the group of addition of the remainders modulo 10?

A: Given: The elements in the group of the addition of the remainders modulo 10.

Q: Write the permutation corresponding to R90 in the left regular representationof D4 in cycle form.

A:

Q: Prove that (1234) is not the product of 3-cycles.  Generalize.

A: Given: Given cycle is 1234 we want to prove that 1234 is not product of 3-cycles.

Q: Consider the permutation (1 2 3 4 5 6 \3 4 5 6 1 2 7)* 1. Express as a product of disjoint cycles.…

A: Given Permutation: 12345673456127

Q: Example 3 A play has a cast of 7 actors preparing to make their eurtain call. Use the permutation…

A: A play has cast of 7 actors preparing to make their curtain call.Use the permutation formula to find…

Q: Write a permutation (1 4 5)(1 5 4 3)(2 1 5) as a product of disjoint cycles.

A: Given that the permutation is

Q: Find the least residue of 448 modulo 7

A: Given problem is : Find the least residue of 448 modulo 7

Q: 5. If a list of nine integers form a group under multiplication modulo 91, which integer is missing…

A: Using Cayley table, ⊙91 1 9 16 22 53 74 79 81 29 1 1 9 16 22 53 74 79 81 29 9 9 81 53 16…

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: Show that a permutation with odd order must be an even permutation

A:

Q: Exercise 14.6.11. Suppose a is an an even or odd permutation? n-cycle. How can you tell whether a is

A: EX 14.6.11 : If σ=(a1 a2 a3....an), We need to explain how can we identify whether σ is an even or…

Q: 6. Prove that the product of an even permutation and an odd permutation is odd.

A: Let S be set on which permutations are defined. Let  α=(x1,x2,...,xk) and  σ=(y1,y2,...yn) be two…

Q: 4. Explain why if p is prime then Z, is the only subdomain of Z,

A: To prove:If p is prime, Zp is an subdomain(internal domain).

Q: Find the order of [5] in Z23

A:

Q: Make a lattice diagram of Z modulo 7 excluding 0 under the binary operation of multiplication

A: Given: ℤ modulo 7 with respect to multiplication. To find: Lattice diagram of subgroups of given…

Q: Find the number of inversions in the permutation 52134 of Sz = {1,2, 3, 4, 5}. 6 O 5

A: Number of inversions

Q: WIKS the largest order of an element in the groupof permutations of 4 objects!

A: What is the largest order of an element in the group of permutations of 4 objects

Q: Which of the following numbers could be the exact number ofelements of order 21 in a group: 21600,…

A: According to theorem, the number of elements of order n in a group is multiple of ϕ(n)

Q: Assume that n > 2 and x E A is a fixed element. How many unordered selections without repeats of k…

A: N>2, X is the element A is a fixed element  How many unordered selections withouts repeats of  k…

Q: How many proper colorings of C15 with color set {1, 2, 3} are there? Hint: find a recursive formula…

A:

Q: (a) How many arrangements of the letters in COMBINATORICS have no con- secutive vowels? (b) In how…

A:

Q: Determine the number of permutations of 2 members of a group from a group with 6 members.

A: Given n=6 r=2

Q: 4. In how many ways can a committee of 5 memebers be chosen from a group of 10 memebrs?

A: Given that, The number of members in the committee is 5. And the total number of members is 10. So,…

Q: Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about…

A: The proof of third part is long. So, if you need the proof, you may re-post the particular question…

Q: The order of the element (1,1) + ((4,4)) in the factor group Z, × Z,/((4,4)) is 6. Select one: True…

A: Given statement is, " The order of the element 1,1+4,4 in the factor group  ℤ6×ℤ84,4 is 6. " The…

Q: 9. Use the algorithm to generate the next 22 permutations of {1,2,3,4,5} starting with 13275.

A: Given: A permuation {1, 2, 3, 4, 5} The algorithm for generating permutation follows an increasing…

Q: Suppose that " is in lowest terms and that m< n. Then has a simple-periodic dec- imal representation…

A: Introduction: The ratio of given two numbers could be represented as a factor less than one always…

Q: 13.How many permutations are there of the digits in the string 12345 that leave 3 fixed but leave no…

A: Solution is given below:

Q: Calculate the least residues modulo 5 of 14, 24, 34, and 44.

A:

Q: - Let 1 2 3 4 5 6 7 8 2 3 5 6 7 1 8 4 1 2 3 4 5 6 7 8 4 8 3 6 1 2 5 7 and be permutations on the set…

A:

Q: Suppose that you want to choose a (2k + 1) − element subset of the n − element set (1, 2, ⋯ , k ).…

A:

Q: What is the largest order of an element in the group of permutations of 4 objects?

A:

Q: Suppose n > 3 is an integer. Prove that in Sn every even permutation is a product of cycles of…

A:

Q: Suppose that p is the smallest prime that divides |G|. Show that anysubgroup of index p in G is…

A:

Q: Find the number of conjugacy classes of S4 and the number of elements in each of these classes

A: Here, S4 is the permutation group of order 24.Number of elements in S4 = {(1),(1 2),(1 3),(1 4),(2…

Q: 5. Suppose we have an enumeration of A ao, a1, A2, ·.. (1) without repetitions (i.e., all the a; are…

A:

Q: Show that the permutation [n, n-1, ... , 2, 1] has n(n-1) inversions.

A: An inversion in a permutation is a pair of numbers such that the larger number appears to the left…

Q: Find the integer a, if the modular representation of a is (2, 1, 5) with respect to the modulo 3,…

A: We have to find the integer a if the modular representation of a is 2,1,5 with respect to modulo 3,…

Q: I. Consider the permutation (1 2 3)(4 5)(1 25 4)-2. 1. Express as a product of disjoint cycles. 2.…

A: consider  the permutation (123)(45)(1254)-2

Q: Suppose that for each prime p, Zp is the homomorphic image of agroup G. What can we say about |G|?…

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: 7. Find a gcd of 49,349 and 15,555 in Z.

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.