# Show the range of f is isomorphic to a quotient of z

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1. Apr 30, 2015

### HaLAA

1. The problem statement, all variables and given/known data
Let G be any group and a in G, define f: Z → G by f(n) = a^n

Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z

2. Relevant equations

3. The attempt at a solution
The range of f is a^n , then quotient group of Z is Z/nZ
Apply the first isomorphism theorem , we have a^n isomorphic with Z/nZ

2. Apr 30, 2015

### fourier jr

if you can show that f is a homomorphism & find its kernel then you'll have your isomorphism by the first isomorphism theorem. the range of f is actually {an ∈ G | n ∈ Z}, not just an. the kernel of f is {m ∈ Z | am = e}, not nZ (but you're not far off). you might need to show that (Z, +) is a group in order to make sure that f is actually a group homomorphism, unless you've already established that in your class.

Last edited: Apr 30, 2015