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

Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: Give all the possible elementary divisors of a group of order 40.

A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5

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: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

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

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

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Q: What is the smallest positive integer n such that there are three nonisomorphicAbelian groups of…

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Q: If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal,…

A: Given: The order of group is 15

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

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

Q: Every cyclic group or order n is isomorphic to (Zn, +n) and every infinite cycle group is isomorphic…

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Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: Every cyclic group is a non-abelian group. True or False they why

A: We have to check

Q: many

A: We have to find the number of generator of the given group of order

Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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Q: Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of…

A: Definition of subgroup: Let (G ,*) be a group and H be a subset of G then H is said to be subgroup…

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

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

Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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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: Give an example or explain why the following is not possible.  An infinite group that is finitely…

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

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Q: Every commutative group has at least element ??

A: Every commutative group has at least               element ?  We know that , every commutative group…

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: The operation of composing any two symmetries in a dihedral group is commutative.

A: To find: The given statement true or false. The given statement: "The operation of composing any two…

Q: Give a list of all groups of order 8 and show why they are not isomorphic. for this you can show…

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

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Q: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

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: Prove that a factor group of a cyclic group is cyclic

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Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.

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Q: Suppose that <a>, <b> and <c> are cyclic groups of orders 6, 8, and 20,…

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Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

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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: 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: The group (Z6,6) contains only 4 subgroups

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Q: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

Q: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: Every finite group is cyclic

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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: 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: Is the set of numbers described below a group under the given operation? Rational numbers;…

A: This is a question of abstract algebra.

Q: The group (Z6,+6) contains only 4 subgroups

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Q: Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.

A: A subset H of G is called a subgroup of G if H also form a group under the same operation.

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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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: Find the order of each element of the group Z/12Z under addition

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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…

Q: Give the Cayley table for the group Z2 under multiplication modulo 12.

A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…

Q: 2) Given example of an infinite group in which every nontrivial subgroup is infinite.

A: Let G=a be an infinite cyclic group generated by a, whose identity element is e. Let g∈G, g≠e,…

Q: G1 = Z÷ and G2 = Z, *

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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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