Qi/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*) Q2/ In (Z9, +9) find the cyclic subgroup generated by 1, 2,5

Q: 16

A: It is known that in the group Zn , if  d divides n , then Zn has a unique subgroup of order d and…

Q: 8. Find a non-trivial normal subgroup of the octic group. Demonstrate that this subgroup is normal.

A: According to the given information, it is required to find a non-trivial normal subgroup of the…

Q: IL Show thai al subgroup of erder 125. 18. Show that there are no simple groups of order 255 =…

A: By using syllow theorems the solution is given as follows :

Q: 1. Show that the symmetry group of F is a subgroup of the isometry group in the Euclidean plane by…

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Describe all the elements in the cyclic subgroup of generated by the 2×2 matrix     [1  1 0  1]

A: Describe the elements in the subgroup of the General linear group GLn, ℝ generated by the 2×2…

Q: (a) G = Z12 and H = { 0, 2, 4, 6, 8, 10 } (based from the operation table) (b) G = Z; and H = { 1,…

A: Given a G=Z12 and H=0,,2,4,6,8,10b G=Z7* and H=1,2,4c G=Z* and H=a+b2 : a,b∈Q and a,b are not both…

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

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Q: 6.10 Find all subgroups of Z,XZ4. 6.11 Find all subgroups of Z,xZ,>

A: To find  All subgroups of z2*z4

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

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Q: Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external…

A: We know the following theorem. Theorem: Aut(Zn) is isomorphic to the group of multiplicative units…

Q: Find all proper nontrivial subgroups of Z, x Z, x Z. Find all subgroups of Z; x Z4 of order 4.

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Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

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Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: To solve this problem,we ues two results 1)if a belongs to H then Ha=H 2)INDEX: the cardinality of…

Q: Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12…

A: Let D2n = {e, y, y2 …… yn+1}{s, sr,…. Srn} .y = rotation .s = reflection Then {e, y, y2, …… yn+1} is…

Q: Please help me understand the following question and please explain the steps. Picture is below

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Q: True or false? The union of any two subgroups of a group G is a subgroup of G.

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Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G G and is…

A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e  also x6=e therefore…

Q: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

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Q: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

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Q: (3) If H={0,6,12,18}, show that (H,+24) is a cyclic subgroupof (Z4,+24). Also list the elements of…

A: Subgroup

Q: (1) Z/12Z

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Q: Find the number of G-orbits in X := {1,2,3, 4, 5, 6, 7, 8, 9, 10}, where G < S10 is the subgroup…

A: Solution:Let g1 = (1,2), g2 = (3,4,5) - and g3 = (6,7,8,9). All three cycles at sight are…

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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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&gt;1 has trivial center

Q: a.) Show that if G is solvable and N≤G is a subgroup then N is solvable b) Show that if G is…

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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: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

Q: Example: Show that (Z,+) is a semi-group with identity

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Q: Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external…

A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

Q: G2 is abelian if and only if G1 is cyclic. The set of all even integers 2Z is a subgroup of (Z, +)…

A: Given that 2Z is the set of all even integers.

Q: Theorem group tren G'45. Eremple Treorem is i0 9 N a group then Example abeliar.

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Q: The centralizer and normalizer of a subgroup of a group are the same . If thats true give deatiled…

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Q: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

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Q: (a)If G = {-1,1}, then show that (G, .) is an abelian group of order 2, where the operation…

A: G={-1,1} be a set of two elements.

Q: Exercis 1 Show that every group of order 45 has a normal subgroup of order 9.

A: We have to prove that group of order 45 has a normal subgroup of order 9

Q: ve Lagrange's theorem and apply it to find all subgroups of the group 7.12.

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Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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Q: your a) G = (Z12,O); H = {0,2,4, 6, 8, 10} %3D

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Q: {1,2, 3} under multiplication modulo 4 is not a group. {1,2, 3, 4} under multiplication modulo 5 is…

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Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: Q1)) Prove or disprove (only two) : 1) Every subgroup of P-group is P-group.

A: The solution are next step is

Q: If N is a normal subgroup of G and G/N is abelian. Then G is also abelian. Select one: True False

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Q: An is a * normal subgroup of Sn None of the choices not a subgroup of Sn subgroup of Sn but not…

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Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: True or False If every proper (other than the group itself) subgroup of a group is cyclic then the…

A: Note: According to our guidelines We’ll answer the 3 questions since the exact one wasn’t specified.…

Q: 21. Define Alternating group. Can you explain this concept (order, is it subgroup of Sn)?

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

A: We need to show that (G,*) is cyclic.

Q: Let 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

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