December 11th, 2010, 06:52 AM  #1 
Newbie Joined: Nov 2010 Posts: 7 Thanks: 0  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 
December 11th, 2010, 08:39 PM  #2 
Senior Member Joined: Oct 2009 Posts: 105 Thanks: 0  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. 
December 12th, 2010, 06:34 AM  #3 
Newbie Joined: Nov 2010 Posts: 7 Thanks: 0  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? 
December 20th, 2010, 03:03 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2207 
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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