I read this link . In Theorem $2.7 $, it is mentioned that for $n\geq 3$ except for $n = 5, 6, 8$, symmetric group $S_n$ is generated by an element of order $2$ and an element of order $3$. However, we also know that for $n\geq 2$, $S_n$ is generated by the transposition $(1 2)$ and the $n$-cycle $(12\ldots n)$. If we use later result then $S_4$ is generated by transposition $(1 2)$ and $4$-cycle $(1234)$ which contradicts result of Theorem $2. 7$ since order of four cycle is four.
Could anybody explain me where I am going wrong? I would be very much grateful.
Thanks for your time.