1. ## Homomorphisms/Isomorphisms

a) The inclusion Z -> R is a homomorphisms of additive groups.

b) The subgroup {0} of Z is isomorphic to the subgroup {(1)} of S5.

These are both true statements, but I must know how to prove them. Can someone show me please? Thanks.

2. ## Re: Homomorphisms/Isomorphisms

Originally Posted by jzellt
a) The inclusion Z -> R is a homomorphisms of additive groups.

b) The subgroup {0} of Z is isomorphic to the subgroup {(1)} of S5.

These are both true statements, but I must know how to prove them. Can someone show me please? Thanks.
Let me ask you a question, if $\displaystyle \iota:\mathbb{Z}\hookrightarrow \mathbb{R}$ is our inclusion what is $\displaystyle \iota(x+y)$? What about $\displaystyle \iota(x)+\iota(y)$?

3. ## Re: Homomorphisms/Isomorphisms

I'm not sure. If f is our function f: Z -> R, doesn't it have to be defined for me to continue?

4. ## Re: Homomorphisms/Isomorphisms

Originally Posted by jzellt
I'm not sure. If f is our function f: Z -> R, doesn't it have to be defined for me to continue?
We have defined the function, it's the inclusion map. Let's pause for a second, did you take the time to fully understand all the words in the problem statement? Do you know what an inclusion map is?

5. ## Re: Homomorphisms/Isomorphisms

nevermind see next post...

6. ## Re: Homomorphisms/Isomorphisms

So I googled it and it looks like the inclusion map of function f is defined by f(x) = x. So to show that f: Z->R is an homomorphism, I must show that f(x+y) = f(x) + f(y). But f(x+y) = x+y and f(x) + f(y) = x+y. So were done. COrrect?

7. ## Re: Homomorphisms/Isomorphisms

Originally Posted by jzellt
So I googled it and it looks like the inclusion map of function f is defined by f(x) = x. So to show that f: Z->R is an homomorphism, I must show that f(x+y) = f(x) + f(y). But f(x+y) = x+y and f(x) + f(y) = x+y. So were done. COrrect?
Exactly.