Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a generator of the group.

Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

Q: Suppose that G is a finite abelian group.  Prove that G has order pn where p is prime, if and only…

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Q: Let G be a cyclic group with more than two elements: 1. Prove that G has at least two different…

A: Given: Let G be a cyclic group.

Q: Prove that every group of order 330 is not simple.

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Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

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: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

Q: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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Q: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that a group of even order must have an element of order 2.

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Q: Every quotient group of a non-abelian group is non-abelian.

A: (e) False (f) True (g) True Hello. Since your question has multiple sub-parts, we will solve first…

Q: Prove that a group of order 12 must have an element of order 2.

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Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: prove that any group R=3 must beperiedio

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

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

Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of the order…

A: Here given that G is a finite Abelian group that has exactly one subgroup for each divisor of the…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: Prove that every subgroup of an abelian group is abelian

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Q: Let G be a cyclic group of order n. Let m < n be a positive integer. How many subgroups of order m…

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Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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

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Q: 10. Prove that any cyclic group is abelian.

A: As you are asked multiple questions but as per our guideline we can solve only one. Please repost…

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A:

Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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Q: = Prove that, there is no simple group of order 200.

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

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Q: Show that a finite group of even order that has a cyclic Sylow 2-subgroup is not simple

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Q: Find the number of isomorphism classes of the abelian groups with order 16.

A: The order of abelian groups = 16

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: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

Q: Characterize those integers n such that the only Abelian groups oforder n are cyclic.

A: According to the question,

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: List all abelian groups (up to isomorphism) of order 600

A: To List all abelian groups (up to isomorphism) of order 600

Q: Let Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

Q: - Iet G be a non-trivial group with no non-trivial proper subgroup. Prove that G is a group of prime…

A: Let G be a non-trivial group with no non-trivial proper subgroup. We need to prove that G is a group…

Q: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

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Q: What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups…

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Q: Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: (iv) Does there exist a group G such that [G, G] is non-abelian? Give an example, or prove that such…

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Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of |G|. Show…

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Q: suppose H is cyclic group. The order of H is prime. Prove that the group of automorphism of H is…

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