2- Let (C,) be the group of non-zero -complex number and let H = {1,-1, i, -1}. Show that (H,;) is a subgroup of a group (C,). %3D

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Dn Prove that is isomorphic to a subgroup of Sn

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Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

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Q: Prove that The set of all complex number is a group under multiplication, and G={1,-1,i,-i}  is a…

A: Hi! You have posted multiple questions. As per norms, we will be answering only one question and as…

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: Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⊕ Z2 but…

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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: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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Q: Exercises: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y? = 2}.…

A: Subgroup of a group

Q: Exercise: Show that (Z5, +5) is an abelian group.

A: The element of Z5={0,1,2,3,4}

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

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Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: 3. Define Lie group.

A:

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: Example: H.W 1- Let (C\{0},.) be a group of non-zero complex number and let H = {a + ib, a? + b2 =…

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

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

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: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

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Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

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: Show that 1 y? ry under a Lie group corresponding to the symmetry generator X = r² a + ry

A: Given,

Q: 1+2n Prove that if (Q-(0},) is a group, and H = a n, m e Z} 1+2m is a subset of Q-{0}, then prove…

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Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

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Q: 1. Show that G is closed under x. 2. Show that (G. x) in a cyclic grouP generated by t.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: True or false? The group S3 under function composition ◦ is not a cyclic group

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Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

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Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

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

A:

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

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Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

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

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

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Q: ii) Find the structure of its Galois group, G.

A: To Determine :- The structure of its Galois group, G.

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: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: Q2 : Find the left regular representation of the group Z5 and express the group element in the…

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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: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

Q: Find the order of the element (2, 3) in the direct product group Z4 × 28. Compute the exponent and…

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Q: - Iet G be a non-trivial group with no non-trivial proper subgroup. Prove that G is a group of prime…

A: Let G be a non-trivial group with no non-trivial proper subgroup. We need to prove that G is a group…

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

A:

Q: 3. Show that Q has no subgroup isomorphic to Z2 × Z2.

A: The objective is to show that ℚ has no subgroup isomorphic to ℤ2×ℤ2

Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

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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: Q2// Let Hi family of subgroups of (G, *). Prove that the intersection of Hi is also * .subgroup

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Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

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