list all group of order 1 and 12, then 1. list all elements of the groups and their order. 2. determine whether the groups are cyclic. 3. determine whether the groups are abelian. 4. determine all non-trivial proper subgroups and normal subgroups of the groups. 5. list down the group(s) that is/are isomorphic to the group Use definitions/theorems to support the work.

Q: abouis 13. Let (G, *) be cyclic group of finite order n and let a € G. Prove that ak is a generator…

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Q: group correspoindire by a² and if ~1 = {a} find the quotient to subgroups generated respectively.

A: Given that G={a} is a cyclic group of order 8.Then find the quotient groups corresponding to the…

Q: 4. Let p and q be distinct prime numbers and let n = p²q². Determine all the subgroups of Zn and…

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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: 5. Find the number of generators of the cyclic group Z15

A: To find the number of generators of the cyclic group ℤ15.

Q: prove that a group of order 45 is abelian.

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Q: prove and that NOXMALIZER STABLIZERS of CENTRALIZERS" are Sub group 67 groupt

A: To prove that normalizer is a subgroup. The normalizer is defined as N(H)=x∈G:x-1Hx=H. Take, x=e,…

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: (This is the 19. Let G be a group, and let H be a subgroup with [G: H] = 8. If g E G has odd order…

A: Bezouts theorem will be used in solving this question.

Q: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: For a cyclic group of order 420, the number of generators are, (a) 66 (b) 76 (c) 86 (d) 96

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Q: (b). Let N = {1,2,3,...,}. Let G = Sym(N) be the group of all bijec- tive functions from N→ N. Then…

A: This is based on simple calcuations of functions. We have used the notations as used for working in…

Q: 4. Calculate the order of the element 4 in group (Z*18, x mod 18) 5. Calculate the order of element…

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Q: Fix g e G, where G is a group whose (original) operation we will suppress in the notation, for…

A: Group Properties: 1) a*b∈G 2) (a*b)*c=a*(b*c) 3) There exists an identity element e such that…

Q: 4.6. (a) Let G be a cyclic group of order 6. How many of its elements generate G? Answer the same…

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Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…

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Q: Q1 Let g be an element of a group G such that r = g and r² = e. Then solve for r in terms of g. Q 2…

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Q: a The group is isomorphic to what familiar group? What if Z is replaced by R?

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Q: 4. The group defined by the following table is called the group of quaternions. Use the table to…

A: NOTE:- According to the company guidelines we are supposed to answer only first 3 sub-parts. Kindly…

Q: Classify groups of order 2p as best as you can. Give a proof for your assertions and, when…

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Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: List all of the subgroups of the group (Z20. +). empty subset of G tha

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Q: c) How many subgroups does (Z36, O) have? What are they? 5.5 Find all the subgroups of Qs. Show that…

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Q: 1. Which of the following statements are true and which are false ? Give reasons for your answer:…

A: To Determine :- Reason for True or False and give reasons.

Q: Let G be a group with less than 50 elements having subgroups of orders 4 and 10. What can you…

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Q: 1. Show that G is closed under x. 2. Show that (G. x) in a cyclic grouP generated by t.

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

Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose <G, .> is a group then G is an abelian if and only if…

Q: is a group with identity (eg, eH).

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Q: Determine whether the following statements are true or false. Justify your answer with brief…

A: To determine : Whether the given statements are true or false and justify the answers.

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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Q: Que 327: Let X, Y be cyclic groups of order 2,3 respectively. Find the orders of the elements of…

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

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Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: (True or False; Prove or disprove) 1. The order of any subgroup of a finite group divides the order…

A: 1. The order of any subgroup of a finite group divides the order of the group.  This statement is…

Q: The n-th dihedral group Dn is the group of symmetries on a regular n-gon. The group of symmetries on…

A: Given : The n-th dihedral group Dn is the group of symmetries on a regular n-gon. The group of…

Q: (7.1) Let (G,) be a group. Prove that the right cancellation law holds in G. Provide reasons in your…

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

Q: Intersection of subgroups is O a. cyclic group O b. group O c. subgroup O d. not subgroup e. abelian…

A: Intersection of subgroups is also a subgroup.

Q: 3. consider the group D4 = {(1), (1234), (13), (24), (1432), (12), (34), (14), (23), (13), (24)} a.…

A: To enumerate all the distinct subgroups of the group D₄, we need to consider all possible…

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: 2. Find two distinct proper subgroups of the additive group (Z12, +}.

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Q: 5. Consider the "clock arithmetic" group (Z15,0) a) Using Lagrange's Theorem, state all possible…

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Q: QI/AI Prove that the mathematical system (s), where (o) is an usual composition map is a group? B/…

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