# Math Help - S_n generated by two elements

1. ## S_n generated by two elements

Hello all , i have tried to show it with no progress .

Can someone help me with that one.

Thanks a lot .

2. The transposition $\sigma = (1\;2)$ and the cycle $\tau = (1\;2\;3\;\ldots\; n)$ generate $S_n$. In fact $\tau^{k-1}\sigma\tau^{-k+1} = (k\;k{+}1)$, so all transpositions of consecutive numbers are in the subgroup generated by $\sigma$ and $\tau$. Then any transposition can be expressed as a product of those, for example $(1\;3) = (1\;2)(2\;3)(1\;2)$. Finally, it is well known that the transpositions generate the whole of $S_n$.