Prove that the order of a k-cycle in S, is k.

Q: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

Q: Express each permutation as a product of disjoint cycles and find the orbits of each permutation

A: Given: 1234535241

Q: An example of a partition of the integers is E Z}

A: Set of integers is defined as {... -3, -2, -1, 0, 1, 2, 3,....}. A partition of a set is defined as…

Q: Prove that every group of order 330 is not simple.

A:

Q: Let G be a graph of order n and let k be an integer with 1≤k≤n−1.Prove that if every vertice of G…

A:

Q: Prove that a group of order 7is cyclic.

A: Solution:-

Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: Determine the subgroup lattice for Zp where p is any prime number

A:

Q: Suppose that Φ is a homomorphism from D12 onto D3. What isΦ(R180)?

A:

Q: Label eachof the following statement/s as either true or false.   Every permutation can be expressed…

A: Every permutation can be expressed as a product of disjoint cycles.

Q: How many elements of a cyclic group with order 10 have order dividing 5?

A:

Q: cyclic

A:

Q: Find a noncyclic subgroup of order 4 in U(40).

A: Let U(40) be a group. Definition of U(n): The set U(n) is set of all positive integer less than n…

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Find all subgroups of U(7)

A:

Q: Give two reasons why the set of odd permutations in Sn is not a subgroup.

A: To show that, The set of odd permutations in Sn is not a subgroup. A subset H of the group G is a…

Q: Every group of order 4 is cyclic. True or False then why

A: Solution

Q: Prove that the following graph does not contain a Hamilton cycle. a b с d h g e m k

A:

Q: Prove that A5 is the only subgroup of S5 of order 60.

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: 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: Find the orders of each of the elements of U(14). Show that it is cyclic and find all its…

A: Given :-  The orders of each of the elements of U14. To Determine :- It is cyclic and all its…

Q: 2. Use the Greedy and Edge-Picking algorithms to find the Hamiltonian cycle with the least total…

A: Circuit: EBCFDE Length: 177 for any doubt   cutting the number we can't Select because 3rd edge…

Q: Let H x K = { (h,k) |h in H, k in K} such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: here ℤ2=0,1

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

A:

Q: How many elements of a cyclic group with order 14 have order 7?

A:

Q: 5: (A) Prove that, every group of prime order is cyclic.

A:

Q: Express each permutation as a product of disjoint cycles and find the orbits of each permutation…

A: To express the permutation 1, 4, 8, 71, 9, 6, 51, 5, 3, 2, 9 as a product of disjoint cycles…

Q: Express each permutation as a product of disjoint cycles and find the orbits of each permutation

A: Given that, Let p = 1  2  3  4  5   6  73  4  5  6  1   2  7  be a permutation Two cycles are said…

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

Q: List all of the cycles in S4.

A: The symmetric group Sn is the group of bijections from any set of n objects, which we usually just…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

A:

Q: Find a subgroup of order 4 in U(1000).

A:

Q: Prove if n greater than or equal 2 and G is connected with 2n odd vertices then G has n open walks…

A: Hamiltonian Path: If we remove any one edge from a Hamiltonian circuit. then we left with a path.…

Q: Express each permutation as a product of disjoint cycles and find the orbits of each permutation

A: In this permutation  1→12→3→24→5→4

Q: Show that if T is an involution, and S is any other transformation, then STS⁻¹ is an involution

A:

Q: Let f and g be disjoint cycles in Sn. Prove that fg = gf

A: Given: Let f and g be disjoint cycles in Sn. To Prove that: fg = gf

Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: (5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e G.

A:

Q: Express each permutation as a product of disjoint cycles and find the orbits of each permutation

A: a)1234523415b)1234513542

Q: 1. Prove or give a counter example to the following conjecture: If G and G both contain an Eularian…

A: We have to prove or give a counter example to the following conjecture: If G and G¯ both contain an…

Q: Find all the generators tof the subgroup H = (2) in Z24-

A: In any cyclic group of order n has phi(n) generators. We use this technique to solve the problem.…

Q: Label eachof the following statement/s as either true or false. The mutually disjoint cycles of a…

A: Solution: Consider the given statement is: The mutually disjoint cycles of a permutation are the…

Q: Let G be an odd cycle. Prove that any proper 3-coloring of G using the colors red, green, and blue…

A:

Q: .

A: To prove : The two cycles (a1,a2,...ak), (b1,b2,...bm) are disjoint cycles. If the pair of cycles ↵…

Q: 14. If σ is a cycle in Sn of length 5, then σ^2 is also a cycle. Select one: True False

A: Given statement

Q: a. Show that in any group of 50 or more people there are at least 5 of them with the same birth…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Find the cycle of decomposition of o.

A: Given: σ=(1 2 3 )(4 5 )(7 6 1) To find: 1) The cycle of decomposition σ. 2)The σ-1. 3) Express σ as…

Q: Show that a group of order 77 is cyclic.

A: