Find the kernels and images of the following homomorphisms. Which of the homomorphisms are injective? Which are surjective? (i) The mapping 0 : Z₁₁ → Z₁ given by 0(x) = x². (ii) The mapping ☀ : R* → R* given by o(x) = x³. (iii) The mapping : G → G given by y(x) = g-¹xg for x E G where G is a group and g € G. (You may assume that this is a homomorphism.)

Q: (d) Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-"(x) # -VIn x?

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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: Which are invertible

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Q: Prove the following property about subdifferential, affline transformation. - əf(x) = {Vf(x)} if ƒ…

A: We will solve both the questions for you and present the solution in detail. We only need the basic…

Q: Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

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

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A: We see,         [2] = [1] + [1]    but, f([2]) = ([0], [1])          f([1]) + f([1]) = ([1], [1]) +…

Q: The endomorphism A:Rn[x]->Rn[x] acts according to the rule A-> p - p(0). Then the rank of the…

A: Step 1:na Step 2: Step 3: Step 4:

Q: (3) Conjugate the map f(x) = 2x² +x – 3 to a map of the form Q(x) = x² + c. Find the conjugacy map h…

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Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

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Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

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A: Well ordering

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Q: Let zo + 1 be a zero of order 3 of an analytic function g(z) and f (z) = (z-1)2 (g(z))³" Then z, is…

A: Use the concept that if f(z) has zero at z0 of order m then 1/(f(z)) has pole at z0 of order m. By…

Q: φ : F(α) → F(α) such that: φ(α) = (aα+b)/(cα+d), and φ is the identity map on F. (1) Show that the…

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