Exercise 3: Prove that every element of a finite group is of a finite order.

Q: Exercise 3.2.6 Show that if G and H are isomorphic groups, then G commutative implies H is…

A: A group G is called Commutative if for any a,b in G imply ab=ba

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: Exercise 7.2.9. Suppose G is a group and H, K ≤ G finite subgroups such that gcd (|H|, |K|) 1. Prove…

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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: Prove that there is no simple group of order 300 = 22 . 3 . 52.

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Q: Theorem 3-12 Every infinite cyclic group is isomorphic to (Z,+). Proof.

A: Explanation of the solution is given below...

Q: Construct an element of multiplicative group of the finite field elements.

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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: 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: 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: 5. Define the right regular action of a group G on itself. Show it is a group action. Is it…

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Q: Prove that any two groups of order 3 are isomorphic.

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Q: Find an isomorphism from the group G = to the multiplicative group {1, i, – 1, – i} in Example 3 of…

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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: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

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: What is a quotient group and conjugacy class

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Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

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Q: Prove that the set of natural numbers N form a group under the operation of multiplication.

A: The set N of all natural numbers 1, 2, 3, 4, 5... does not form a group with respect to…

Q: Determine all elements of finite order in R*, the group of nonzeroreal numbers under multiplication

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Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: The prime factors of 525 are 3, 5 and 7. So there are proper normal subgroups of order either 3,5 or…

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: 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.

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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: 5: (A) Prove that, every group of prime order is cyclic.

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Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: Suppose that o is a homomorphism from a finite group G onto G and that G has an element of order 8.…

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

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Q: suppose G is Finite group and FiGgH homogeneity : prove that Ir(6)|iG| Be a

A: Given that G is a finite group and F:G→H is a homogeneity, i.e. F is a homomorphism. To prove  FG |…

Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Suppose that G is a finite group and that Z10 is a homomorphicimage of G. What can we say about |G|?…

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Q: Theorem: Any non-commutative group has at least six elements

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Q: 26. Show that any finite subgroup of the multiplicative group of a field is cyclic.

A: We use the fundamental theorem of finite abelian group.

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: Let (G,*) be a finite group of prime order then (G,*) is a cyclic

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Q: Prove that cent (G ) is cyclic group G is commutative

A: If cent(G) is cyclic group, then G is commutative If G is commutative, then cent(G) is cyclic group

Q: Show that if G and H are isomorphic group, then G commutative implies H is commutative also.

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

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

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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: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.

Q: 10. Prove that all finite groups of order two are isomorphic.

A: Here we use basic definitions of Group Theory .

Q: Prove that in any group, an element and its inverse have the same order.

A: Proof:Let x be a element in a group and x−1 be its inverse.Assume o(x) = m and o(x−1) = n.It is…

Q: Give an example of a finite group that is not abelian.

A: We have to give an example of a finite group that is not abelian. Pre-requisite : A non-abelian…