a) Is there any relation between the automorphism of the group and group of permutations? If exists, give example.

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

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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: Is the set of integers a commutative group under the operation of addition? Yes; it satisfies the…

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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: 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: Show that the group of permutations Σ2 is abelian. Then show that Σ3 is not. Writing up the group…

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Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

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

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

Q: 22, Use mathematical induction to prove that if a1, a2, ... , an are elements of a group G, then…

A: See the detailed solution below.

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

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Q: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: Let F denote the set of first 8 fibanacci number .Then convert F into a non abelian group by showing…

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Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

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

A: Yes , every subgroup of z is cyclic

Q: b. Find all abelian groups, up to isomorphism, of order 720.

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

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Q: Suppose that G is an Abelian group with an odd number of elements.Show that the product of all of…

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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 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: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

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

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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: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

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

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Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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

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Q: 3. Prove or disprove: If G is a group, (g¬')-' = g.

A: Consider the given information: Let G is a group. To show that (g-1)-1=g

Q: Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group…

A: To verify corollary to the Fundamental Theorem Of Finite Abelian Groups Where, G is a group of order…

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

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

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

Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

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

A: Solution

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: True or False: (a) Two finite non-cyclic groups are isomorphic if they have the same order. (b)Let o…

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Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b - a for all a…

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Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

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: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

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Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

Q: (5) Make a list of all group homomorphisms from Z4 → Z8.

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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.…