This is quite embarassing but I've been revising an algebra text and I fail to get past through a supposedly easy detail.
Take the symmetric group of permutations $S_{3}$ and its two elements: $(1 2 3)$ and $(1 3 2)$. Now any multiplication table tells you that $(1 2 3) \circ (1 2 3) = (1 3 2)$ and $(1 3 2) \circ (1 3 2) = (1 2 3)$. I just can't figure out why it is true.
For example, $(1 2 3) \circ (1 2 3)$: we first send $1$ to $2$, then $2$ to $3$ by the $2^{nd}$ permutation. Secondly, we send $2$ to $3$ and then $3$ to $1$, which leads to $(1 2 3) \circ (1 2 3) = (2 3 1) = (1 2 3)$. What am I doing wrong? Did I misunderstand the notation?
Thank you.