1. ## Isomorphism

Let G be the group of all real-valued functions on the unit interval [0,1], where we define for f,g elements of G, addition by (f + g)(x) = f(x) + g(x) for every x elements of [0,1]. If N = { f element of G,f(1/4) = 0 }, prove that G/N is ismorphic to real numbers under addition

2. Originally Posted by Godisgood
Let G be the group of all real-valued functions on the unit interval [0,1], where we define for f,g elements of G, addition by (f + g)(x) = f(x) + g(x) for every x elements of [0,1]. If N = { f element of G,f(1/4) = 0 }, prove that G/N is ismorphic to real numbers under addition

What about H: G --> R+ defined by H(f):= f(1/4)? Can you prove this is a homom. of groups and it is onto? Then use the first isomorphism theorem.

Tonio

3. Originally Posted by tonio
What about H: G --> R+ defined by H(f):= f(1/4)? Can you prove this is a homom. of groups and it is onto? Then use the first isomorphism theorem.

Tonio
Thanks

4. Originally Posted by Godisgood
Thanks but am still not able to prove is homomorphism and onto?Can u pls show me how to dat that..

Hmm....to study this stuff you must have taken before at least an introductory calculuis course and even a little more, and linear algebra and stuff...

Is is true that H(f + g) = H(f) + H(g)? Or what is the same, is it true that (f + g)(1/4) = f(1/4) + g(1/4)? To answer this just check CAREFULLY the definition of addition in G...

About onto: is it true that for any real number r we can find a real valued function defined on [0,1] s.t. f(1/4) = r?

Please do think a little about these questions and make an effort towards their solution, and THEN if you're still stuck write back.

Tonio