elementary transpositions
Consider the product of transpositions $s_3$$s_1$$s_4$$s_3$$s_2$$s_4$$s_1$$s_2$
In my book they go from this to (23)(56)(45)(12)(34)(45)(56) and then to
(1362)
My teacher said you with elementary transpositions you can calculate products much faster than regular because they have algebraic properties but he said to research it and didnt tell me and I cant find on google
How does the product of (23)(56)(45)(12)(34)(45)(56) so quickly and easily come out to (1362)? I understand 4 and 5 are single cycles so ignored
I have struggled with multiplying transpositions in the past myself.
The way you go about it is: start with any number, for example $1$, and write it down:
$$(1$$
Now from right to left, look for a transposition with a $1$. If you find none, then you close the parenthesis:
$$(1) $$
And skip to the next number. If you find one, you replace the number in your head with the number that comes with $1$. For example, in your product $(23)(56)(45)(12)(34)(45)(56)$ a $(12) $ comes up so I store the $2$ in my head and then continue right to left, looking for a $2$. Eventually I find one in $(23) $ so I swap the $2$ in my head with the $3$. Now the sequence ended so I write the $3$:
$$(13$$
And repeat the process with the $3$. I start with it and go right to left. I find a $(34) $ so change to $4$, then a $(45) $ sp change to $5$, and then a $(56) $ so change to $6$. The sequence ends so write the $6$ down:
$$(136$$
And repeat... starting with the $6$ we end with $2$ so I write the $2$
$$(1362$$
Now I will start with $2$ and end with $1$, thus closing this cycle:
$$(1362) $$
Now you pick one of the remaining numbers, $4$ for example:
$$(1362)(4$$
And repeat the process. Starting in $4$ ends in $4$ so
$$(1362)(4) = (1362) $$
And the same thing happens for $5$.