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

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: Prove that a group of order 375 has a subgroup of order 15.

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Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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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: 4. Give an example of a pair of dihedral groups that have the same number of conjugacy classes as…

A: We know that Dihedral group is denoted by Dn. Its order is 2n. Here, to find the pairs of Dihedral…

Q: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

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

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Prove that a group of order 7is cyclic.

A: Solution:-

Q: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

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: Prove O3 is not a group

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Q: · In a group, prove that (ab) = b-'a-|

A: As you asking for question number 7 , I solve for you.

Q: Show that the center of a group of order 60 cannot have order 4.

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Q: (H,*) is called a of (G,*) if (H,*) is a group.

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

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

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

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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: How many nonisomorphic abelian groups of order 80000 are there?

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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: Every group of order 4 is cyclic. True or False then why

A: Solution

Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

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

A: Solution:-

Q: Prove that a group of order 15 is cyclic

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

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Q: Show that any group of order 4 or less is abelian

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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: The Sylow theorems are significant in the categorization of finite simple groups and are a key…

Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

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

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

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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: A cyclic group is abelian

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Q: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

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: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

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: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

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: Let G be a cyclic group of order 33. List the orders of all its elements and how many elements have…

A: Here, G is a cyclic group of order 33.  We have to find the orders of all its elements and the…

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

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

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

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: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

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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: 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.…