How to find the Permutation in $S_8$ given as the product $C= (1483)(12765)(34687)$?

Question

I don't understand how to answer this. Just by reading it off, "$1$ goes to $4$, and $4$ goes to $6$, thus $1$ goes to $6$" but that logic doesn't match with the answer I've been giving: Answer: $(127)(386)(45)$.

So how do you go about answering this type of question? What logic do you use in reading $C$?

Thanks.

Answer 1

you're going in reverse order. start on the right and work left.

i) 3 -> 4 and 4 -> 8, so 3 -> 8

ii) 8 -> 7 and 7 -> 6, so 8 -> 6

iii) 6 -> 8 and 8 -> 3, so 6 -> 3

together these give the cycle (386)

iv) 4 -> 6 and 6 -> 5, so 4 -> 5

etc

Answer 2

You should read the product of cycles from right to left. The reason why you do this is because the cycles are thought of as functions on your set, so the product is really function composition, which is read from right to left.

Answer 3

Different authors use different notations. It looks like the notation used in your text is that when we write a composition $ghk$ of three permutations, $k$ is the permutation which is applied first, then $h$ and then $g$. So you just need to read those cycles from right to left.