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The question asks me to determine if $4\mathbb{Z}$ and $5\mathbb{Z}$ (with standard addition) are isomorphic and if so to give the isomorphism.

My attempts: What I am having difficulty with is showing a mapping that preserves the operation. i.e., $\phi(a+b) = \phi(a) + \phi(b)$.

What I have so far is:

$\phi(a+b) = \phi(5a/4) + \phi(5b/4)$.

$\phi(a+b) = 5/4(a+b) = 5/4a + 5/4b = \phi(a) + \phi(b)$.

Therefore, my conclusion is $4\mathbb{Z}$ and $5\mathbb{Z}$ are isomorphic and the isomorphism is $\phi(n) = \frac{5}{4}n$.

Can someone either confirm this or point me in the right direction? I really appreciate everyone willingness to help one another! Thank you!

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    Seems fine to me. Since $n$ is divisible by 4, $\frac{5}{4}n\in 5\mathbb{Z}$. You just need to show injectivity and surjectivity.2010-12-14
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    Also, there is only one infinite cyclic group. Each group you listed is an infinite cyclic group.2010-12-14
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    @Sean Tilson: "There is only one infinite cyclic group" *up to isomorphism*.2010-12-14
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    @ Arturo:but of course... (and that isomorphism need not be unique!) (good catch though) @user: you should take my mistake to mean that cook people don't really worry about two things being different when they happen to be isomorphic. Unless they are super cool, then they worry a lot about the choices of isomorphisms.2010-12-14

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