Multiplying permutation cycles

[fishturtle1]

Homework Statement

Compute each of the following products in $S_9$ (Write your answer as a single permutation).

a) (145)(37)(682)

Homework Equations

Theorem: Every permutation is either the identity, a single cycle, or the product of disjoint cycles.

The Attempt at a Solution

I start with (145)(37)(682).

Going from the rightmost cycle to the left, starting with 6.. 6 goes to 8, 8 goes to 8, then 8 goes to 8. So 6 goes to 8.

So i have (68) so far.

Now 8 goes to 2, 2 goes to 2, 2 goes to 2. So 8 goes to 2.

Now i have (682).

Now 2 goes to 6, 6 goes to 6, 6 goes to 6. So 2 goes to 6.

Now i have (6826) and I'm going to have a loop like (682682682...). I'm not sure where to go from here. If I just skip to 7, like (6827), then 2 goes to 7.. but 2 needs to go to 6 somehow. Also, by the theorem, its true that a permutation cannot be expressed as a single cycle and the product of disjoint cycles. Its a product of disjoint cycles..so is what I'm doing impossible?

 

[Orodruin]

Your original product is a product of disjoint cycles. I believe the problem statement parenthesis should be "your answer should only contain disjoint cycles". It is unclear what "single permutation" means.

 

[fishturtle1]

Your original product is a product of disjoint cycles. I believe the problem statement parenthesis should be "your answer should only contain disjoint cycles". It is unclear what "single permutation" means.

I asked here first to see if I could get a hint before looking at the answer key. It seems "single permutation" means the form:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 & 4 & 5 & 6 \\ \end{pmatrix}$

I can take it from here, sorry for the weird question, I was trying to avoid the answer key on the very first question..

 

[Orodruin]

Ok, that makes sense. I just don't like that notation as it takes more space and is less transparent than cycle notation, but as a first exercise to get the hang of what it means I guess it is fine.