Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the group of real numbers with the operation of addition and that R* = R\{0} is the group of real numbers without 0 and with the operation of multiplication. ¡. Let ƒ₁ : R → R be defined by f₁(x) = -2x for all x E R. ii. Let ƒ₂ : R → R be defined by f2(x) = |x| for all x ¤ R iii. Let ƒ3 : R* → R* be defined by f3(x) = |x| for all x ER* iv. Let G be the subgroup of GL₂(R) given by G-{(61) a+0,DER} (ö |a = Let f4 GR be defined by f5 : a (81) = b.

Q: You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for a longel…

A: Given that You are buying a car. Would you prefer a higher rate for a lower term or a lower rate for…

Q: Let G:- [0, 1) be the set of real numbers x with 0<x<1. Define an operation + on G by ** y={x+y if…

A: Solution:-  Given   G=0,1   be the set of real numbers xwith  0≤x&lt;1.  The operator ∗on G is given…

Q: In the frieze group F7, write x-3zxyz in the form xnymz k.

A: It is known that F7=xnymzk∥n∈Z,m=0,k=0 or k=1 and x is the translation of R2, and it map u,v to…

Q: 15. Let G be a group and let g be in G. Define a function :G → G by ,(x) = gxg Show that :G→G is…

A: Let G be a group and let g be in G . Define a function ϕg : G→G by ϕgx=gxg-1. To prove: ϕg : G→G is…

Q: 4. Let G be a group and g e G. Prove that the function f: G G given by f(x) = gx is a bijection.

A: The solution for the asked part , is given as

Q: 5. Let a E G be fixed. Define y: G → G by p(x) = axa-1 for all x E G. Prove that p is a…

A: To find - Let a∈G be fixed. Define ψ:G→G by ψx=axa-1 for all x∈G. Prove that ψ is a homomorphism.

Q: Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab…

A: Let A be an associated algebra.  To prove that for any a∈A, the map da:A→A defined by dab=ab-ba is a…

Q: 16. Let R* be the multiplicative group of non zero real numbers and let S= the group under the…

A: As per the guidelines I have solved first problem.

Q: Let T be the transformation defined by the formula Then, the .T X2 x3 x3 codomain is اخترأحد…

A: The codomain is ℝ2 Hence, option A is correct.

Q: R\{-1} by a * b = a + b+ ab. Show that (R \ {-1},*) is an 3. Define the operation abelian group. *…

A: The solution for the asked part , is given as

Q: Suppose that f: XY is a homomorphism (of either groups or rings). Prove that f is injective if and…

A:

Q: Justify if it is true: If T:V → W is a monomorphism, then V = T(V)

A:

Q: Prove that the mapping from U(16) to itself given by x → x3 is anautomorphism.

A: To show that the mapping from U16 to itself given by x→x3 is an automorphism, we need to show that…

Q: Is the transformation H: C( [0, 1] ) → C( [0, 1] ), H(f) = e* f(x) a. linear?? b. one-to-one?? c.…

A: A mapping  T : U to V is said to be linear  if for x,y inU and k is scalar , then…

Q: Given groups (R+, ) and (R, +). Show that maping ƒ: (R, +) → (R+,-) ex X Vx E (R, +) is an…

A: Given  that   and   are  groups.Let  be  defined  by  1) is  well-defined  and  onre-to-one: Thus, f…

Q: unctions are well-defined group homomorphisms? Se elections; you cannot score less than 0 on this…

A:

Q: Let f R2 R² be an isomorphism where : → ƒ ( [¹³]) = [()] and ƒ ([×4]) = }}] Find(G)

A:

Q: 2. If a is an involution, then Baß is an involution for any transformation B.

A: It is given that α is an involution. So, we can say that α2=I.

Q: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: Let f: X → Y and g : Y → Z be arbitrary maps of sets (a) Show that if f and g are injective then so…

A:

Q: — 2х21 Show that T: R² → R² defined by T()= - 15.) is an X2 isomorphism. X1

A: T is a n isomorphism

Q: A $ 100 deposit is made into a college fund at the beginning of each month for 10 years. The fund…

A: Consider the given information. The deposit amount is $100. The rate is 9% and the time is 10 years.…

Q: Let G = RX be the group of nonzero real numbers under multiplication. Let r is a positive integer.…

A: Given: G=ℝ* be a group of non-zero real number under multiplication and r is a positive integer. To…

Q: 17. If E is an extension of F and f (x) e F[x] and if is an automorphism of E leaving every element…

A:

Q: a b 2.3 Let e: (M,(R), +) → be defined by e a + d. where a,b,c,d ER (a) Prove that e is a…

A: Given that θ:M2ℝ,+→ℝ,+ is a mapping defined by θabcd=a+d where a, b, c, d∈ℝ. We have to prove that θ…

Q: How can I prove that f is one-to-one (injective)?

A: We know, a function/map f is one to one (injective) iff          For a ≠ b, f(a) ≠ f(b) for any two…

Q: Find a rule for the following isomorphism: 9: Z2 x Z3 + Zz x Z2 p (a, b) = (

A: we have φ:z2×z3→z3×z2 we know a result: A G1⊕G2 is isomorphic to G2⊕G1 by g1,g2→g2,g1.

Q: If α and β are transformations on a set S, prove that α-1 ∘ β-1 and β-1 ∘ α-1 are transformations.…

A: Linear transformation are represented as the matrix . If A and B are two matrix then (AB)-1=B-1A-1…

Q: Suppose (X,≺) is a well-ordering and f : X → X is an isomorphism. Then f is the identity function on…

A: Well ordering

Q: . Prove that the shift o is a homeomorphism.

A: To, prove that σ is homomorphism we will show that the inverse image of open set is open. So, the…

Q: Let R* denotes the set of positive real numbers and let f: R+ R+ be the bijection defined by ƒ (x) =…

A: In this solution, we will examine the properties of the binary operation ⊗ on the set of positive…

Q: Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective…

A:

Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: ✓1.20 Show that these are homomorphisms. Are they inverse to each other? (a) d/dx: P3 → P₂ given by…

A:

Q: Let A B and g: B-C be maps. Prove that if g of is one-to-one, then / is one-to-one.

A:

Q: 1. Let G be the set of integers under addition and H = {-1,1} be a multi- plicative group. Show that…

A:

Q: Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. this option f: (R, +) →…

A:

Q: Let G (R,+). Show that f: GG given by f(x) = 2x is an isomorphism.

A:

Q: (b) Let G=(Z8, +) and f: GG be defined by f(x) = 2x and show that f is a homomorphism that is…

A: