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Let G denote the set G = {f : R → R | f is inﬁnitely diﬀerentiable at every point x ∈ R}. (R as in the reals)

(a) Prove that G is a group under addition. Is G a group under multiplication? Why or why not?

(b) Consider the function ϕ : G → G deﬁned by ϕ(f) = f′

Prove that ϕ is a homomorphism with respect to the group operation of addition. What is the kernel of ϕ?

(c) Consider the function ψ : G → G deﬁned by ψ(f) = f′′ − f. Prove that ψ is a homomorphism with respect to the group operation of addition. What is the kernel of ψ?

My work:

For (a) we need to prove that G satisfies all the group axioms under addition. Let f, g, h be infinitely differentiable at every point x in R

Closure : f + g is infinitely differentiable at every point x in R

Associativity : (f + g) + h = f + (g + h)

Identity : Find a function e in G such that for all functions f in G such that f + e = e + f = f

Inverse : For all f in G find an inverse ~f in G such that f + ~f = e

Is G a group under multiplication?

For (b) and (c)

The kernel of a homomophism are all elements of the domain group that map to the zero element, e, of the codomain.

What is the zero element of the codomain? This is e I found for G in the group axioms. In this problem e is the zero function zero(x) = 0?

So for (b), the problem is asking for which functions f in G does phi(f) = zero, i.e. which functions has f'(x) = 0 for all x in R?

for (c), the problem is asking for which functions f in G does psi(f) = zero, i.e. which functions has f''(x) - f(x) = 0 for all x in R?