1. Classification of a factor group: a. Determine the order of (3,2) in Z₁8 × Z12 18 b. Determine the order of (Z18 × Z12)/((3,2)) c. Classify (Z18 × Z12)/((3,2)) according to the fundamental theorem of finitely generated abelian groups

Q: In the group Z24, let H =(4) and N= (6). (a) State the Second Isomorphism Theorem. (b) List the…

A: As per our guidelines only first three subquestions are solved. To get solution of remaining…

Q: Prove that the Galois groups of x10 -1 and x 8 -1 over Q are notisomorphic.

A: Given: Galois group of x10-1 and x8-1. To Prove: Galois group of x10-1 and x8-1 over Q are not…

Q: give the order of the element in the in the factor group (2,5) + in (Z3 x Z6)/

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Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

Q: If (G,X) be the group of all real 2 x 2 matrices (b) such that ad - bc #0 with matrix multiplication…

A: Given if G, × be the group of all 2×2 matrices abcd such that ad - dc≠0 with matrix multiplication…

Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

Q: O In the group G=Z%6 , let H (9) and K = (18) %3D List the elements inG / H , G/K and in H/K ,…

A: For G/H : H=0,9,18,271+H ={1,10,19,28}..........8+H={8,17,26,35} For G/K:…

Q: Let G - { (a, b) : a eR, be R\{0} }. Define on G the binary operation: (a, b) (c , d)- (a + bc , bd)…

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Q: (a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where is defined by (a, b) * (c, d) = (ac,…

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

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: Let (G, ') denote the set of all 2 x 2 real matrices A with det{. and det {A} € Q (the rational…

A: a)  Let A, B ∈ ( G, ·) ⇒ det {A} ≠ 0, det {A} ∈ QAlso det {B} ≠ 0, det {B} ∈ QNow det {AB}= det {A}…

Q: Prove that there exist no non-abelian simple groups of order < 60.

A: Let G be a group such that G&lt;60 and G is non abelian. To Prove, the group is non Simple. Since…

Q: O(G) If G is a finite group, than o(G) = Σ- D(N(a)) where this sum is taken over one element of each…

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Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

Q: Find the factor group G divides H where G is the quaternions,H=Z(G)

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Q: b. Prove Theorem 3.16: Let a be an element of a group G. The centralizer of a in G is a subgroup of…

A: See the detailed solution below.

Q: 4 -1 Let A be the matrix A = 2 1 1 250 1 1 50 If we express A 1 as then find a, b, and A. O 50 a b O…

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Q: a)Let G be a group of order pq, where p and q are primes. Prove that either G is abelian or Z(G) =…

A: Given that G is a group of order pq , where p and q are primes.Z(G) We have to show that G is…

Q: Consider the groups U(8) and Z4: (1) Determine the identity element in the group U(8) ×Za: (ii)…

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Q: Determine the Galois group of x2 +9 over R.

A: The objective of the question is to determine the Galois group of x2+9 over R.

Q: Show that the Galois group of a polynomial of degree n has orderdividing n!

A: Let us consider that the polynomial, fx=∑i=0ncixin, ci∈F Also let K be the splitting field of fx…

Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

A: Since , we have to find order of the factor group U(16)/&lt;9&gt; Concept: Order of factor group…

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

A: It is given that,  GL(n, ℝ) is the set of general linear group of all n×n matrices and  SL(n, ℝ) is…

Q: (a) List all the finite abelian groups (upto isomorphism) of order 1800.

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Q: b. Show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image.

A: To show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image, we need…

Q: Find an isomorphism o from the multiplicative group G of nonzero complex numbers to the…

A: We need to find ϕ for the multiplicative group G of non zero complex numbers to the multiplicative…

Q: . Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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Q: (4) (a) State the fundamental theorem of cyclic groups. (b) State the fundamental theorem of finite…

A: In first we State the statment of fundamental theorem of cyclic groups. Secondly we State that  the…

Q: (2) (Z x Z)/(6Z × 14Z)

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Q: Show that x is a generator of the cyclic group (Z3[x]/&lt;x3 + 2x + 1&gt;*.

A: Given: The cyclic group Z3x/x3+2x+1*We need to show that x is a generator of the given cyclic group.

Q: If (G,X) be the group of all real 2 x 2 matrices (“ ") such that ad - be + 0 with matrix…

A: First we have to show that f:G→G' is homomorphism or not. To show homomorphism, we have to show…

Q: 3. Let D be an integral domain. (a) Prove that FD is an abelian group under the operation of…

A: Let us prove that field of fractions is an abeliean group under addition.

Q: (а) (Q, +) (b) (Zs, ·)

A: (a)(ℚ,+)(b)(ℤ8,.)

Q: State Fermat's Little Theorem. In the groun (7/1017)x find 5100 5101 and 5202 (101 is a prime…

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Q: Show that x is a generator of the cyclic group (Z3[x]/&lt;x3 + 2x + 1&gt;)*.

A: In this question, we have to show that x is a generator of the cyclic group  ℤ3xx3+2x+1*. We know…

Q: Prove that (Z × Z)/((0,1)) is an infinite cyclic group. Prove that (Z × Z)/((1,1)) is an infinite…

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Q: Let G = R – {-1}. Define the binary operation * on G by x * y = x +y+xy. a) Show that (G, *) is an…

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