Problem 6. Suppose that X = (X,*) is a group containing at least two elements. Prove the following result. ● If the group X = (X,*) is cyclic, then exactly one of the following statements is true. 1. The group X is isomorphic to the group Z. 2. The group X is isomorphic to the group Zn for some fixed integer n > 1. You did most of the work for this proof in Investigation 8. Problem 6 tells us that, up to isomorphism, there are only two types of cyclic groups those that behave like the integers under addition, and those that behave like the integers under addition modulo n.

Q: 2.4 Prove theorems 1. Identity elements of a group is unique. 2. Every element of a group has unique…

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Q: Problem 2. Let f be a homomorphism from a group G into a group H. Prove that f is one to one if and…

A: Let f be a homomorphism from group G into group H. Suppose f is one to one . We need to show that ,…

Q: 1. Is * commutative? or Is * associative? 2. Compute a*(b*c). 3. Compute (a*b)*c.

A: This is a problem of Group Theory.

Q: 3. Prove that the groups (R.) and (R,+) are not isomorphic, where (R* R{0},-) is the multiplicative…

A: We have to show that the groups ℝ, + and ℝ*, · are not isomorphic. Here, the group ℝ*, · does not…

Q: 1. Show that the set {5, 15, 25, 35] is a group under modulo 40. What is the identity element of…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: If G is a group then Z(G)◄G. *True False

A: Use the relevant definitions.

Q: 11. Consider the group D, For the subgroup H = {(1), (12)(34)}, write all cosets that can be formed.…

A: D4=Dihedral group of order 4H=1,1234=R0,V and H=2To form the cosets.

Q: 30. Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba…

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Q: 2.11. The group S3 consists of the following six distinct elements e, σ, σ’, τ, στ, σετ, where e is…

A: As per the guidelines I am suppose to solve only three subparts of the given problems.

Q: 3. Infinite, abelian (and additive), and so that for every element a, a+a=0. (Hint: Think of…

A: Note the question asks us for an example of an infinite group such that sum of two elements is…

Q: (c) Consider the subgroups H = (−1) and K = (5) of G. Show that GHxKZ₂ x Z2m-2. (d) As an…

A: Lot of number theory facts will be used to solve this problem.

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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Q: Problem 34: Let G be a group. о(а) Show that o(a") = for all a e G %3D (о(а), п) where n is an…

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Q: this problem, consider the group G = S3 acting on X = S4 by conjugation. other words, given an…

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Q: 10. (a) If gh hg in a group and o(g) and o(h) are finite, show that o(gh) is finite.

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Q: 7. (a) What is the order of (1, (123)) in Z4 × S3 ? (b) Write down all the distinct cosets of the…

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Q: 8. Let S = (12345), T = (13)(45) in S5. Describe the group generated by S and T?

A: Sol:- The group generated by S and T is the group {S, T} = ⟨S, T⟩. We first observe that S and T…

Q: 4

A: We use theorem of order of group and order of elements of group .

Q: Problem 6 Assume x, y are elements of some group G, |x| = n and [y] = m. Suppose that x and y…

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Q: 9. Here's the Klein group K again: * RO R2 H V RO RO R2 H V R2 R2 RO V H H HV RO R2 V VH R2 RO…

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Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

Q: Question 1. Let G = Q be the set of all rational numbers, and * = + be the usual addition operation…

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: Let the cosets of the subgroup e in problem 3 be written as N=0+N and g+N, where g is 1. What is…

A: Let the cosets of the subgroup e in problem 3 be written as N=0+N and g+N, where g is 1. What is…

Q: 1. Z30 2. U(14)

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Q: Problem 1. Find all subgroups of the group Zand draw the lattice diagram for the sub- groups.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q1)…

Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: 7. Let (A,*) be a semigroup that satisfies the property VaEA, a*A=A and A★a=A Show that A is a…

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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Q: 2*. Let Q/Z be the group described in problem 12 of Worksheet 1.1. Find list the elements of the…

A: To identify the subgroups generated by the given elements in the quotient group Q/Z.

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

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Q: 1. Let M2(R) = {C )la,b,c, d eR and ad – bc + 0}. You already know . 아 that M2(R) forms a group…

A: A

Q: Problem 5. (a) What is the order of the dihedral group D6? How many elements of order 2 are there in…

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Q: 1. This is an exercise from math100a which gives us a characterization of cyelic groups. (a) Suppose…

A: It is given that

Q: 2. Finite, abelian (and additive), not isomorphic to Z₂ or Z2 x Z2, and also for every element a,…

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Q: 4. Let S5 denote the symmetric group on the 5 letter set > let be elements of S5. -1 - (92234)…

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Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…