I'm trying to compute (12)(1253)
What I Did:
I started with 1, and 1 goes to 2 and 2 goes to one, so (12).
Then I did 5, which goes to 3. 3 goes to 1 and 1 goes to 2. Then 2 goes to 5, so it's back around again.
What I Got:
(12)(532)
Is this correct?
So we have $1 \to 2$ on (1 2 5 3). We then see on (1 2) that $2 \to 1$. So we have $(1)$ as our first cycle. We only write down a number if we start with it, or if we don't see it to the left of our current position.
Now start at $2$, so open $(2$. Note $2 \to 5$. We don't see $5$ again, so write down $5$: (2 5. Now start at $5$, which goes to $3$. We don't see $3$ again, so write down $3$: (2 5 3. Now start at $3$. We see $3 \to 1$, and then $1 \to 2$. We would write down $2$, except we started at $2$, so close the cycle. So we have (1)(2 5 3) = (2 5 3) as our answer.