1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding isomorphism

  1. Nov 17, 2004 #1
    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 dont know what Im 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...
  2. jcsd
  3. Nov 17, 2004 #2
    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)...
  4. Nov 17, 2004 #3

    I understand by reading the book what all the steps ask me to do, but I dont 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 dont know what the hint "try n to 2n" means. How am I supposed to use that....

    sorry, really newbie at this.
  5. Nov 18, 2004 #4
    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...
  6. Nov 18, 2004 #5


    User Avatar
    Science Advisor
    Homework Helper

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

    What you have to check now is:
    Injectivity: [itex]f(x)=f(y) \Rightarrow x=y[/itex]
    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).
  7. Nov 18, 2004 #6
    I think that whats 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...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Finding isomorphism
  1. Isomorphic math help (Replies: 6)

  2. Finding acceleration (Replies: 18)