Decompose permutation $σ=(123)(345)(4567).$

Question

i'm getting confused at how to decompose this permutation I've seen this post about how to decompose How to decompose permutations?

I understood and managed to apply to other kind of permutation but I'm struggling with this one

σ=(123)(345)(4567)

σ(1) = 2 because 1->2 in (123)

That's where i'm confused :

σ(2) = 5

2 -> 3 , 3 -> 4 ( in 345), 4-> 5 (in 4567)

Do I stop here ? Or continue 5-> 3 ( in 345), 3->4 (in 345), 4->5 ( in 4567) but it's getting weird, i'm entering a loop when do I actually stop ?

Answer 1

You're right: $\sigma(2)=5$ (when read left-to-right).

You don't just stop there, although that's it for how $\sigma$ behaves on $2$. The next number to consider is indeed $5$.

Once you hit a number you've considered before, move on to the next number you have not (if there be such).

You should get $$\sigma=(1253)(467).$$