Products of permutation cycles

Question

Hey guys im having trouble understanding how to calculate products of permutation cycles, especially when its several 2 or 3 cycles in a row, I have a few short questions here with the answers, but I can't understand how to arrive at these answers.

1. (56)(45)(34)(12)(23)(12) Answer: (16543) (2)
2. (45)(34)(45)(34)(56)(45)(56) Answer: (1)(2)(3465)
3. (123)(234)(324) Answer: (123)(4)
4. (213)(324)(324) Answer: (134)(2)

I know how things work with products of 2 cycles A and B but I can't figure out how to apply it to repeated cycles or more than 2 products. Could someone please assist if possible? would be appreciated :)

Answer 1

Start with the number 1, write that down and proceed right-to-left through the permutations, which may or may not be single cycles. For each encountered permutation, work out the image of the number in hand and "mutate" it to the image. Write the final image down, and repeat the right-to-left process with the final image until reaching the initial number again.

This is one cycle of the composed permutation. Repeat the whole process of the last paragraph, starting with the first number not yet written down, to get the other cycles.

For the first example:

(56)(45)(34)(12)(23)(12)
 6<  5<  4<  ..  3<  2< 1
 5<  ..  ..  ..  ..  .. 6
 ..  4<  ..  ..  ..  .. 5 (16543)
 ..  ..  3<  ..  ..  .. 4
 ..  ..  ..  1<  2<  .. 3
 ..  ..  ..  2<  ..  1< 2 (2)

Answer 2

Apply the permutations successively. Start from the right and move to the left. Just record what happens to each of $1,\dots,6$.

For instance, $(16543)$ can be expressed by the following "word": $621345$. This means: $1\to6,2\to2,3\to1,4\to3,5\to4$ and $6\to5$.