I am unsure the formal mathematical terminology/notation for dealing with sequences generated from integer modulo arithmetic. So first off, could someone recommend a book that focuses on the sequences generated from arithmetic operations on finite sets of integers? I bought an elementary number theory book and an abstract algebra book but neither ever discussed sequences.
Now the more explicit question. Consider: $f \left[ n \right] = a n \left( \text{mod} \; N \right) \\ g\left[n\right] = b n \left( \text{mod} \; N \right) $ where $a,b \in \mathbb{Z}$, $n \in \{0, 1, 2, \ldots \}$, $N \in \{2, 3, 4, \ldots\}$ and $a \left( \text{mod} \; N \right)$ is the remainder of $a / N$. I want to prove that sometimes the sequence $g[n]$ "generated" (may not be correct term) by a choice for $b$ is a "permutation" (may not be correct term) of the sequence $f[n]$ generated by $a$ if $N$ is the same both. I also want to show that the new permutation can be generated by $g \left[ n \right] = f \left[ k n \right]$ for some $k \in \mathbb{Z}$ whenever such a permutation exists. I have a suspicion that if $a$ and $b$ are relatively prime to $N$ then such a permutation exists from what little I know of congruence relations, but I can also think of cases where $a$ is relatively prime to $N$ and $b$ is not that this still works.