Let $e$ denote the identity permutation, $\sigma = (12)$, and $\tau = (13)$.
I understand that $\langle \sigma \rangle$ would be taking powers of $\sigma$. So $\langle \sigma \rangle$ would give me $(12), (12)(12) = (1)(2) = e.$
Now I want to consider $\langle (12),(13) \rangle$. From above $(12)^{2} = e$, and $(13)^{2} = e$. Then $(12)(13) = (132).$ Also $(12)(12)(13) = (12)(132) = (13).$
My question is I could never get $(13)(12) = (123)$ right? And have I got all possible products generated by $(12),(13)$?
Thanks in advance.