Cycle decomposition of given action I don't understand the second part of this question at all could somebody explain to me what they want me to do for the second part?
Let $S_3$ act on the set $\Omega$ of ordered pairs: $\{(i,j) : 1 \leq i,j \leq 3 \}$ by $\sigma((i,j)) = (\sigma(i),\sigma(j))$. Find the orbits of $S_3 \ on \ \Omega$. For each $\sigma \in S_3$ find the cycle decomposition of $\sigma$ under this action.
Now I solved first part by brute force,however I don't understand the second part what do they want me to do ? 

I'll given an example. Denote
$$a=(1,1), \quad b=(1,2), \quad c=(1,3), \\ d=(2,1), \quad e=(2,2), \quad f=(2,3), \\ g=(3,1), \quad h=(3,2), \quad i=(3,3). $$
Consider the transposition $\sigma=(12)\in S_3$. We have
$$ \sigma(a)=e, \quad \sigma(b)=d, \quad \sigma(c)=f, \\ \sigma(d)=b, \quad \sigma(e)=a, \quad \sigma(f)=c, \\ \sigma(g)=h, \quad \sigma(h)=g, \quad \sigma(i)=i. $$
Therefore, with $\sigma$ acting as a permutation of $\{a,b,c,d,e,f,g,h,i\}$, its cycle decomposition is
$$(ae)(bd)(cf)(gh)(i). $$
Now do the other $\sigma$s in $S_3$.
