Prove that $\langle \{(123…n),(12) \} \rangle =S_n$

Question

I know that $\langle \{(123…n),(12) \}\rangle \subset S_n$. I was thinking that if I could show that $\langle \{(123…n),(12)\} \rangle $ contains all the transpositions of $S_n$ then it would contain $S_n$. How would I go about showing this?

Answer 1

You can first prove that $S_n$ is generated by $(k\;\;\;k+1)$ for $k=1,2,\ldots,n-1$. Then, check that $$(1\;2\;3\;\ldots\;n)^{k-1}(1\;2)(1\;2\;3\;\ldots\;n)^{-(k-1)}=(k\;\;\;k+1).$$