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.
Best Answer
The theorem doesn't claim that $S_n$ is only generated by a 2-cycle and a 3-cycle. It claims that it can be generated by these.
The fact that $S_4$ can also be generated by a 2-cycle and a 4-cycle is not a contradiction. Note that $S_4$ can very well be generated by a $2$-cycle and a $3$-cycle: Take, e.g., $\langle (1, 2), (2, 3, 4)\rangle$.