Construct the distinct left cosets of the subgroup {(1), (1,3,2), (1,2,3)} in S3.

Q: Prove that the mapping f: G- G defined by f(x) = x-l is an isomorphism of G onto itself.

A:

Q: - a. Find the derivative of 3y – 2xy² = sin x + cot y b. Find the derivative of 2y x(3³) – In(²/²) =…

A:

Q: 6. Use the intermediate value theorem to show that the function has a real zero between the two…

A:

Q: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

A:

Q: Show that any group of order 3 is abelian.

A: The solution is given as

Q: dz Suppose z is a function of x and y, and tan vy? + x² = z*e®v. Solve for dz and dy

A: The given function is                                  tan y2+x2=zxe6y

Q: Consider the parallelepiped with adjacent edges u = 8i + 4j + k v = i+j+9k w = i+ 9j + 4k Find the…

A:

Q: Suppose a and ß are transpositions, then show that if aß= (1), then a = B.

A:

Q: Prove that if x is a group element then |x| = |x-1|.

A:

Q: COS x sin x Prove that the mapping 0 from R under ordinary addition to GL(2,R) defined by 0(x) = -…

A:

Q: Use Theorem 13.9 to find the directional derivative of the function at P in the direction of PQ.…

A: Directional derivative of the function

Q: Suppose o:Z50 - Z15 is a homomorphism with Þ(7) = 6. (i) Determine p(x) where x € Z50. Hint: Use the…

A: Given that ϕ:ℤ50→ℤ15 is a homomorphism with ϕ7=6. We have to determine ϕx where x∈ℤ50. We know that…

Q: (b) Set up a definite integral that is equal t length of the portion of the parabola y which serves…

A: Introduction: The washer method is used to calculate the volume enclosed by two functions. In this…

Q: Prove that the function f: R -→ R defined by f(x)= x³ is bijective.

A: If a function is both one-one and onto, then the function is bijective.

Q: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: Suppose 0: G-→ G be a homomorphism of groups. Show that: (i) Kere is normal in G (ii) The mappping…

A: Normal subgroup: A subgroup N of a group G is said to be a normal subgroup of G if  gNg-1=N , for…

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

A:

Q: VI. Use Lagrange Multipliers to solve the following: Maximize f(x,y, z) = 4.x +2y + z subject to r…

A:

Q: Let f : A B be function. Prove that the relation a - b if and only if f(a) = f(b) is an equivalence…

A:

Q: Construct the group table for U (24). Find the order of each element in U(24).

A:

Q: Let G = Ja, b ER\ {0}} under the operation of matrix multiplication. Prove or disprove that G is a…

A:

Q: A(2, 4) and B(2,-2) are two points in a rectangular coordinate plane. A moving point P(x, y) lies on…

A: a)  Given that Co-ordinate of the point A is    and Co-ordinates of the point B is      Suppose that…

Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

A:

Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

A:

Q: II. Consider the function g defined by 1 g(x, y) = cos (Tx Vy)+ log3(x – y)' Do as indicated. 1.…

A: First we change the base if logarithm to e.

Q: Let G be a finite group and let K < H< G. Show that [G: K] = [G: H][H: K].

A: The following definitions and theorem are used to prove the required result. The number of elements…

Q: Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G,…

A:

Q: Let S be a subset of Sn where S = {a € Snla(1) = 1}. Show that S is a subgroup of Sn.

A: In order to prove S is a subgroup of Sn, prove that α∘β∈S such that α,β∈Sn and α-1 ∈S .

Q: Let G be a finite group. Show that there exist a positive integer m such that for each gE G, gm = e.

A:

Q: Show your step-by-step solution and answer.

A: We will show the required result.

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.