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December 11th, 2010, 06:52 AM   #1
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The Kernel...

Let G denote the set G = {f : R ? R | f is in?nitely di?erentiable 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 ? : G ? G de?ned by ?(f) = f?
Prove that ? is a homomorphism with respect to the group operation of addition. What is the kernel of ??
(c) Consider the function ? : G ? G de?ned by ?(f) = f
?? ? f. Prove that ? is a homomorphism with respect to the group operation of addition. What is the kernel of ??

I'm having trouble with the kernel stuff. This was in the book and I feel that if I saw this example layed out I could apply it to others I'm having trouble with
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December 11th, 2010, 08:39 PM   #2
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Re: The Kernel...

The kernel is defined to be the set of elements that map to the identity element, which in this case is f(x) = 0 for all x. So consider the ? case, when does the derivative of a function equal 0 (also considering the original restrictions you have set, which I don't think should be a problem).

Let me know if there is something else.
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December 12th, 2010, 06:34 AM   #3
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Re: The Kernel...

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?

How do I complete this?
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December 20th, 2010, 03:03 PM   #4
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From where did you copy and paste the problem?
Do you know of any function f(x) for which f''(x) = f(x)?
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