Solving Computations w/ Cycles: Permutations, Inverses, Orders

[CoachBryan]

So I've been sitting here for a while looking at my study guide and I am not sure how to find the product (or even the inverse) of this permutation in S9:

(2 5 1 3 6 4) (8 5 6)(1 9) = (1 3 8 5 9)(2 6 4) (Correct answer)

I know it starts off with 1 --> 3, then you get (1 3 and then after you continue with 3 --> 6 --> 8. After you start with 6 --> 4, right? But I keep coming across the wrong answer. I get (1 3 8 4 2) (8 1)

I'm pretty confused on this topic. If you could shed some light I'd greatly appreciate it. Thanks!

 

[Erland]

Well, it was a long time ago I did those things, but, assuming that the permutations should be performed from left to right, I get:

1 --> 3 --> 3 --> 3
3 --> 6 --> 8 --> 8
8 --> 8 --> 5 --> 5
5 --> 1 --> 1 --> 9
9 --> 9 --> 9 --> 1

leading to the cycle (1 3 8 5 9)

2 --> 5 --> 6 --> 6
6 --> 4 --> 4 --> 4
4 --> 2 --> 2 --> 2

leading to the cycle (2 6 4)

7 --> 7 --> 7 --> 7

leading to the cycle (7)

So, the answer is (1 3 5 8 9)(2 6 4)(7)

but (7) needs perhaps not be included, or maybe the elements to be permuted don't even include 7. In any case, the book's answer is correct.

 

[CoachBryan]

Well, it was a long time ago I did those things, but, assuming that the permutations should be performed from left to right, I get:

1 --> 3 --> 3 --> 3
3 --> 6 --> 8 --> 8
8 --> 8 --> 5 --> 5
5 --> 1 --> 1 --> 9
9 --> 9 --> 9 --> 1

leading to the cycle (1 3 8 5 9)

2 --> 5 --> 6 --> 6
6 --> 4 --> 4 --> 4
4 --> 2 --> 2 --> 2

leading to the cycle (2 6 4)

7 --> 7 --> 7 --> 7

leading to the cycle (7)

So, the answer is (1 3 5 8 9)(2 6 4)(7)

but (7) needs perhaps not be included, or maybe the elements to be permuted don't even include 7. In any case, the book's answer is correct.

Thanks a lot! Now it makes sense. Do you know how to find the inverse?

 

[Erland]

Thanks a lot! Now it makes sense. Do you know how to find the inverse?

Just go backwards!

 

[HallsofIvy]

The convention that I am familiar with is that if a character is not changed by a permutation, it doesn't need to be mentioned.
So your (1 3 5 8 9)(2 6 4)(7) can be written as (1 3 5 8 9)(2 6 4).

 

[CoachBryan]

Thanks guys! On the inverse problem as well. Appreciate it.