I'm slightly confused about the product of the following permutation cycles.
I am given that $s_1 = (1\ 2)$ and $s_2 = (2\ 3)$ where both are generators for the symmetric group $S_3$.
My textbook proceeds by saying that $s_1 s_2 = [3\ 1\ 2]$ and that $s_2 s_1 = [2\ 3\ 1]$, but I can't really see why. To me it looks like the results have been mixed up.
My understanding is that (correct me if I am wrong),
$s_1 s_2 = (1\ 2)\cdot(2\ 3)=(1\ 2)(3)\cdot(1)(2\ 3)=(1\ 2\ 3)$
which on one-line notation is $[2\ 3\ 1]$.
Similarly, I compute the product $s_2 s_1$ as
$s_2 s_1 = (1)(2\ 3)\cdot(1\ 2)(3)= (1\ 3\ 2)=[3\ 1\ 2]$
Clearly, my results are the exact opposites of the textbook's.
What am I missing? At first I thought it was a typo, but Wolfram Alpha agrees with my textbook so there must be something I have misunderstood.
Usually we apply $(s_1s_2)(i)=s_1(s_2(i))$. Hence $(12)(23)$ applied to $1$, is first applying $(23)$, so fixing $1$, and then moving $1$ to $2$ by $(12)$. Similarly, $2$ is moved to $3$ and then fixed. So in this order, $$ (12)(23)=(123). $$