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

[HaLAA]

Homework Statement

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

Homework Equations

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

 

[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 | a^m = 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.