Cycle Notation for a Permutation Group

Question

Can anyone thoroughly explain how you would arrive at this answer? I'm very confused with how you would do this problem.

Answer 1

Let's do $\sigma_2$ for example. Since $\sigma_2(a)=2a \bmod{7}$, it maps $1\bmod{7}$ to $2 \bmod{7}$, $2 \bmod{7}$ to $4\bmod{7}$, and $4\bmod{7}$ to $8\bmod{7}$, which is $1 \bmod{7}$. Hence we have our first cycle $(1 2 4)$. $3$ is the next number not used, so we start with that. Then $\sigma_2$ maps $3 \bmod{7}$ to $6 \bmod{7}$ and $6 \bmod{7}$ to $12 \bmod{7}$. But $12 \bmod{7}$ is $5 \bmod{7}$, and that finally gets mapped to $10 \bmod{7}$, which is $3 \bmod{7}$ again. Hence we get $(3 6 5)$.