Product Permutations/cycle notation

[wontonsoup]

Homework Statement

This is just an example in a textbook but I'm completely stumped.
This is dealing with cycles, and product permutations.
So we have
p=(341)(52)
q = (1452)

and we want to compute the product permutation qp

Homework Equations

so qp = (1452) * (341)(25)

The book writes p sends 3 to 4 and q sends 4 to 5 so qp sends 3 to 5.

The Attempt at a Solution

I know that we proceed from right to left on the permutations, and go from left to right inside the cycles.


I realize that in a cycle, say (341) 3 -> 4-> 1-> 3.

Then in this problem of finding "qp" we start with 1, and the right-most permutation. Then from there I am lost. I suppose 1 -> 3 -> ?. after 1 sends to 3 in p, do we then find what 3 would send to in q (except there is no 3 in q)?The answer for this example is (135) but I am completely lost on how to proceed. The book unfortunately isn't very clear on this. Could anybody give me some pointers on how to proceed? Thanks!

 

[tiny-tim]

welcome to pf!

hi wontonsoup! welcome to pf!

so qp = (1452) * (341)(25)
…
… 1 -> 3 -> ?. after 1 sends to 3 in p, do we then find what 3 would send to in q (except there is no 3 in q)?The answer for this example is (135) but I am completely lost on how to proceed.

(135) means that qp sends 1 to 3, qp sends 3 to 5, qp sends 5 to 1

qp sends 1 to 3 because (341) sends 1 to 3, and (1452) leaves 3 alone

qp sends 3 to 5 because (341) sends 3 to 4, and (1452) sends 4 to 5

qp sends 5 to 1 because (25) sends 5 to 2, and (1452) sends 2 to 1 ​