Find Isomorphism: Mapping Ints to Even Ints

[semidevil]

find an isomorphism from from the group of integers under addition to the group of even integers under addition.

I know, very simple question, but I don't know what I am doing here...

the hint in the book says to try n to 2n. I thought of that too, since it specificaly says integers to even integers.

the books says to prove injective, surjective, and phi(x,y) = phi(x) phi(y).

so what do I do? start x = 2y and prove x = y?

I think I'm wronng...

 

[nnnnnnnn]

You have n to 2n so try defining a function that creates the isomorphism:

f(x) = 2x

Once you have that, the rest follows:

injective: f(x) = y & f(x') = y now show that x = x'
surjective: you know that if y is an even integer then it is equal to 2x for some x, where x is an integer...

the last part is showing that f(x+y) = f(x) + f(y)...

 

[semidevil]

You have n to 2n so try defining a function that creates the isomorphism:

f(x) = 2x

Once you have that, the rest follows:

injective: f(x) = y & f(x') = y now show that x = x'
surjective: you know that if y is an even integer then it is equal to 2x for some x, where x is an integer...

the last part is showing that f(x+y) = f(x) + f(y)...

I understand by reading the book what all the steps ask me to do, but I don't know what they mean by "define a map or function." Like, what do I map from what to what?

do I do 2x = 2y, and then go through all the steps? what if they ask to find an isomorphism from integers to odd integers, or something? do I do 3x = 3y?

Basically, I don't know what the hint "try n to 2n" means. How am I supposed to use that...

sorry, really newbie at this.

 

[nnnnnnnn]

I understand by reading the book what all the steps ask me to do, but I don't know what they mean by "define a map or function." Like, what do I map from what to what?

do I do 2x = 2y, and then go through all the steps? what if they ask to find an isomorphism from integers to odd integers, or something? do I do 3x = 3y?

Basically, I don't know what the hint "try n to 2n" means. How am I supposed to use that...

sorry, really newbie at this.

The map or function is f(x) = 2x...

To map to odd integers use f(x) = 2x+1, this is not a group though because it is not closed: 3+3=6...

 

[Galileo]

I understand by reading the book what all the steps ask me to do, but I don't know what they mean by "define a map or function." Like, what do I map from what to what?

The question asks you to find (define) an isomorphism from $\mathbb{Z}$ to
$2\mathbb{Z}$:
$$f:\mathbb{Z} \rightarrow 2\mathbb{Z}$$.
The 'hint' (which basically gives the answer) is: try f(x)=2x.

What you have to check now is:
Injectivity: $f(x)=f(y) \Rightarrow x=y$
Surjectivity: for every even number y there exist an integer x, such that f(x)=y.
Homomorphic property: f(x+y)=f(x)+f(y).

 

[nnnnnnnn]

do I do 2x = 2y

I think that what's confusing you is that both have to have the same number of elements. Since both groups are infinite, it doesn't matter.

Try thinking of it this way: the integers under addition represent how many $2 bills you have and the even integers represent how many $1 bills you have...