Consider the two multiplicative groups (R.) and (R;. .) and let f: (R.) → (R.) be a map defined by f(x) = x² for all x E R. (a) Show that f is a group homomorphism. (b) Compute kerf. (c) Is f one-to-one? Justify your answer. (d) Is fonto? Justify your answer. (e) Is f an isomorphism? Justify your answer. (f) Show that R/{-1,1} = R.

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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.

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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

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Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

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Q: Question 10. Prove in detail that G(T) is a subgroup of G.

A: Here we use one step subgroup test

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Q: Question 3: a. Let (H,+) ≤ (R, +), let K = {2h : h E H}. Prove that (K,x) ≤ (R*,x). (Clarification:…

A: We shall use subgroup test to solve this problem

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…

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