Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element. If G is not cyclic how many elements you need to generate this group. What is the smallest size of a generating set?

Q: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: The following is a Cayley table for a group G. The order of 4 is: 2 3 5 2 3 4 5 3 4 1 2 4 2 1 3 2 3…

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: Give the subgroup diagram for each of the groups: (a) Z24 (b) Z36-

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Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

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Q: If G = is a distant group and f: G → G is an automorph, show that f(a) is a generator of G, i.e. G =

A: We need to show every element of G is some power of f(a)

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

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Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: In a group G,let a,b and ab have order 2.show that ab=ba

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Q: 1ABCD E 1 A D E

A: Commutative group of order 6 is z6 under multiplication.

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

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Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: (a) Let G = {e, a, . .. , aº | a10 = e} be the cyclic group of order 10. For which m, is G = (am)?…

A: As per our guideline we are supposed to answer only first asked  question. Kindly repost other…

Q: Does the Z base 5 form a cyclic group w.r.t. addition modulo 5? Justify. If yes, find all the…

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Q: 1. Construct the multiplication table for the the group Us = {1,a, a², a°, aª} where a = 2ni e 5

A: As per our guidelines we are suppose to answer only one ques. Answer of question 1 is as follows:

Q: Which of the following is a group? O O

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab,…

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Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the…

Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140

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Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

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: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

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Q: For any elements, a and b in a group G, a^(5).b = ba and a^(4) = e, then G is an abelian group.

A: Given a and b in a group G. and  a5b = ba  and  a4 = e To prove:- G is abelian group

Q: Suppose x is an element of a cyclic group of order 15 and x3 = x7 = x°. Determine |x13].

A: According to a theorem in group theory , If G is a finite group and a∈G be an element in the group…

Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

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Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

Q: 1. Let G be a cyclic group of order 6. How many of its elements generate G?

A: Any finite cyclic group of order 'n' has total ϕ(n) number of generators.  where 'ϕ' represents…

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: Let p and q be distinct prime numbers and set n = pq. Find the number of generators of the cyclic…

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Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Let (Z's. ) be the multiplicative group modulo 54. a. Is this group cyclic? How many generators does…

A: (a) Zn is a cyclic group of order n. Here n=54. So, Z54 is a cyclic group. The number of generators…

Q: Is the set of numbers described below a group under the given operation? Even integers; addition O…

A: A set is called group on binary operation if a) It is closed b) It is associative c) It has identity…

Q: Find all the generators of the following cyclic groups: (Z/6Z,+), ((Z/5Z)*, ·), (2Z, +), ((Z/11Z)*,…

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Q: G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c E G. 31. Show that if a? = e…

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Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: List all elements of U(10) and give a multiplication table for the group U(10)

A: Given  that, The group U(10). We have to find the all elements in U(10) and multiplication table for…

Q: Let G be a group and a be an element of this group such that a12 = e. The possible orders of a are:…

A: Option (4)

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…

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Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…

A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…