Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

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: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

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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: 7. Show that 4 is a subgroup of S,

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

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Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

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Q: Which of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?

A: Since 0 does not divides 12.

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

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: How many proper subgroups are there in a cyclic group of order 12?

A: let G be a group of order 12 and let x be the generator of the group. Then the group generated by x,…

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

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Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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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: (3) Show that 2Z is isomorphic to Z. Conclude that a group can be isomorphic to one of its proper…

A: (2ℤ , +) is  isomorphic to (ℤ , +) .              Define f :(ℤ , +) →(2ℤ , +) by…

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

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Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Which among is not a subgroup of a cyclic group of order 12? (a*) (a³) O. O Option 4 (a²) (a*) O-

A: We have to check

Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

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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: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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

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Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

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: Determine the order of (Z ⨁ Z)/&lt;(2, 2)&gt;. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

Q: (i). There is a simple group of order 2021.

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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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: The group (Z6,6) contains only 4 subgroups

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Q: Show that 40Z {40x | * € Z} is a subgroup of the group Z of integers. Note: Z is a group under the…

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Q: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

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Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

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: The group U(14) has: اختر احدى الجابات only 2 subgroups 4 sub groups 7 subgroups 6 sub groups

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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: The Kernal of any group homomorphism is normal subgroup True False

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Q: The group (Z6,+6) contains only 4 subgroups

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

A: See the attachment.

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: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

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

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Q: Show that a group of order 12 cannot have nine elements of order 2.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}

A: We have given a group Z4 ,  and we have to find which of the following sets {0,2},{0,3},{0,1} is a…

Q: Show that a group of order 77 is cyclic.

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