List all elements of the group U(15). Is this group cyclic?

Q: Is the set ℤ+ under addition a group? Prove your answer using the properties of the group. Note:…

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Q: Every subgroups of cyclic group is Select one: a. cyclic and normal b. not cyclic and normal C.…

A: Since you have asked multiple questions in single request.so we will answering only first question.…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: True or False. Every group of order 159 is cyclic.

A: According to the application of the Sylow theorems, it can be stated that: The group, G is not…

Q: Applying what we discussed in cyclic groups, draw the subgroup lattice diagram for Z36 and U(12).

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Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: Let G be a group of order 60. Show that G has exactly four elementsof order 5 or exactly 24 elements…

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Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: Every cyclic group is a non-abelian group.

A: False, all cyclic groups are abelian group. This is true

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

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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: 2. Let G be a group. Pro-

A: Let G be a group .

Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

Q: Determine all cyclic groups that have exactly two generators.

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

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

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

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

Q: Consider the groups  and .

A: We have to consider the groups

Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

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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: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

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: 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: A cyclic group is abelian

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Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: (i). There is a simple group of order 2021.

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Q: 8. Give an example of a group G where the set of all elements that are their own inverses does NOT…

A: Let, G,. is a group. Let, G={1,7,17,2,12,3,13} Let, H be a subgroup of G where H={1,7,17,2,12}

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: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

Q: This is abstract algebra:   Prove that if "a" is the only elemnt of order 2 in a group, then "a"…

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Q: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.

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Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

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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 c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

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Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: Show that a group of order 12 cannot have nine elements of order 2.

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

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: construct a cayley table for the dihedral group of order 5

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Q: G1 = Z÷ and G2 = Z, *

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Q: Consider the alternating group A4. Identify the groups N and A4 /N up to an isomorphism.

A: Consider the alternating group A4. We need to Identify the groups N and A4 /N up to an isomorphism.…

Q: b. Find all the cyclic subgroups of the group ( Z6, +6).

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Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…

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