Solving for Isomorphic Subgroups: F/K and the Additive Group of Functions

[ehrenfest]

[SOLVED] group of functions

Homework Statement

Let F be the additive group of all functions mapping R into R. Let K be the subgroup of F consisting of the constant functions. Find a subgroup of F to which F/K is isomorphic.

Homework Equations

The Attempt at a Solution

I have absolutely no idea how to do this. Am I supposed to use the fundamental homomorphism theorem? Is it the set of all functions such that f(0)=0? How in the world would you prove that?

 

[Dick]

The cosets of the quotient group are {f(x)+C} where C varies over the elements of K (konstants) for a given f(x). Pick a representative of each coset in such a way as to form a group. Picking an f(x) such that f(0)=0 works great. So does picking an f(x) such that f(2)=0. You don't need any big theorems. You just need to understand the problem.

 

[ehrenfest]

Here is how you would use the Fundamental Homomorphism Theorem:

Define phi from F to F by phi(f(x)) = f(x)-f(0). It is clear that the kernel of this function is going to be all of the constant functions. It is also clear that the image of this function will hit all the members of F that are 0 at 0. Then the FHT tells us that F/ker(phi) is isomorphic to phi(F) and we are done.

 

[Dick]

For somebody that had "absolutely no idea how to do this", that's really well done. Keep it up! Note f(x)-f(2) for works for phi as well. I kinda like 2.