
Kernel
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?

This question is mostly about understanding definitions.
For (a) 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? Well, is f^{1} always infinitely differentiable when f is infinitely differentiable? Try f(x) = x
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 you 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?
Similar 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?

The forum rules say that you are expected to make an effort. This can be as simple as writing what the definition of group is applied to this situation. You can also describe problems you are having in more detail.