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: , Let G be a finite Abelian group of order n. Suppose G = {91,92,.. {91, 92, 9n} and let x be an…

A:

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: 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: Let R = R \ {-1} and define the operation ♡ on R by a♡b = ab + a + b Va, b E R. Show that (a) ♡ is a…

A:

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: (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: find the fundamental group of X := {(x, y, z) = R³|(x² + y²) (y² + z²)(x² + z² − 1) = 0}

A:

Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

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

A:

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: please answer letter a only

A:

Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

A:

Q: 2. a) The operation in a finite group can be specified by writing down a table. Write the table for…

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

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: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

A:

Q: Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C)…

A: To solve this problem, we use the defination of group.

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: 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: 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: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

A:

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…

A:

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: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…

Q: (2) (Z x Z)/(6Z × 14Z)

A:

Q: Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) =…

A:

Q: Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>*.

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: Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system…

A:

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…

A:

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

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

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

A:

Q: (b) Define Dn → Z₂ by 0(p) = 0 and 0(op') = 1. Show that is a group ho- momorphism. State its kernel…

A: