(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For each integer n, define [tex]f_{n}[/tex] by [tex]f_{n}(x) = x + n.[/tex] Let [tex]G = {f_{n} : n \in \mathbb{Z}}.[/tex] Prove that G is cyclic, and indicate a generator of G.

2. Relevant equations

None as far as I can tell.

3. The attempt at a solution

Doesn't this require us to find one element of [tex]G[/tex] such that, by applying that element over and over again e.g. [tex]f_n(f_n(...)[/tex] we can produce any element of G? My main problem with this is I don't understand how one could find a way to go from positive to negative elements or vice-versa. For example if we let the generator be [tex]f(x) = x + 1[/tex] how could we generate [tex]f_{-1}(x) = x - 1?[/tex] Or do I misinterpret the definition of G/requirements of a cyclic group?

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# Proving a Group is Cyclic

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