Q2/ In (Z9, +9) find the cyclic subgroup generated by 1,2,5

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 that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

Q: Prove that SL,(R) is a normal subgroup of GL,(R).

A: Let G=GL(n,R) be the general linear group of degree n, that is, the group of all n×n invertible…

Q: Every

A: We will be using sylow's theorems and it's consequences to arrive at the conclusion that statement…

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…

A:

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: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

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Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

A:

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: Explain why every subgroup of Zn under addition is also a subring of Zn.

A: Every subgroup of Zn under addition is also a subring of Zn as it follows the 1) Associative…

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

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

A:

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

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

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

A:

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

A:

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

A:

Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

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: (c) Prove that for every divisor d of n, Zn has a unique subgroup of order d. (Hint: What is the…

A: C) Let k be a subgroup of order d, then k is cyclic and generated by an element of order k =K⊂H…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

A:

Q: Find all generators of the subgroup of Z/60Z with order 12.

A: Generators of the group

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Determine all normal subgroups of Dn of order 2.

A: Dn is generated by two elementsa & bwithan=b2=e , andba=a-1 bthenbak=a-k bBy induction and…

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

A:

Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

Q: Determine the galois group

A:

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

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Q: Find a subgroup of order 4 in U(1000).

A:

Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 1 0…

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Q: Use the left regular representation of the quaternion group Q8 to produce two elements of Sg which…

A: Fix the labelling of Q8 , Take elements 1, 2, 3, 4,  5, 6, 7, 8 are 1, -1, i, -i, j, -j, k, -k…

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

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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: Find a non-trivial, proper normal subgroup of the dihedral group Dn-

A:

Q: Can Z121 have a subgroup of order 20? Explain.

A: Subgroup of order 20

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: a dihedral group Dyz (a,bia'- b°- (ab)`=4>. Consider i) Discuss all the possible Commutators of Dy…

A: Let us first find the conjugacy classes for D4.

Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Find all the generators tof the subgroup H = (2) in Z24-

A: In any cyclic group of order n has phi(n) generators. We use this technique to solve the problem.…

Q: (1) Find all subgroupsof (Zs.+s).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 9. [Ine Z) is a subgroup of GL2(R) under multiplication. a) Prove that H = { b) Show that His…

A: 9. (a) To Prove: H=1n01 | n∈ℤ is a subgroup of GL2ℝ under multiplication.   (b) To Show: H is…

Q: Show that S4(a) has no normal subgroup of order three. (b) has a normal subgroup of order four.

A: To prove that (1) No normal subgroup of order 3 exists in S4 and (2) there does exist a normal…