Originally Posted by

**Jhevon** as for this one. if we are mapping to $\displaystyle \mathbb{Z}_5$ then we should be mapping to equivalence classes mod 5, right?

so you mean $\displaystyle \sigma^i \mapsto [i + 5 \mathbb{Z}]$

the order of $\displaystyle \sigma$ is 8, so that $\displaystyle 0 \le i \le 8$, and then everything repeats for higher values. thus, to find the kernel (which i suppose is what you want), it would be $\displaystyle \{ \sigma^i \mid i + 5 \mathbb{Z} \text{ is a multiple of 5}\}$