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

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: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

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

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

A: Solution:-

Q: 9. Prove that a group of order 3 must be cyclic.

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Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: No. of isomorphic subgroup of group of integers under addition is: -

A: As we know group of integers under addition is (Z,+)

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

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

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

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Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

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Q: Let G be a group of order 60. Show that G has exactly four elementsof order 5 or exactly 24 elements…

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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: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

A: Let G be an abelian group of order 108 Find the number of subgroups of order 3. Prove that, in any…

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 there is no simple group of order 216 = 23 .33.

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

A: Solution:-

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: 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: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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

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

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Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

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

A: The Oder is 24

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

A: According to the question,

Q: Show that the groups Z8xZ20xZ12 and Z120xZ4xZ4 are isomorphic by define a one-one and onto map? what…

A: We will use the basic knowledge of groups and abstract algebra to answer this question.

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

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Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

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

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

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 any group of order 75 can have at most one subgroup of order 25.

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

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

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

A: Here we use basic definitions of Group Theory .