How can you prove that the permutations $(1,2,\ldots,n-1)$ and $(1,2,\ldots,n)$ generate the symmetric group $S_n$? I have found proofs for other generators like $(1,2)$ and $(1,2,\ldots,n)$, but not this one. A good paper about this topic is https://kconrad.math.uconn.edu/blurbs/grouptheory/genset.pdf, but does not contain what I am looking for. Thank you in advance!
Asked
Active
Viewed 276 times
1 Answers
3
Let $\sigma=(1,2,\dots,n)$ and $\tau=(1,2,\dots,n-1)$. Then $\tau^{-1}\sigma$ (where permutations are composed right-to-left) is $(n-1,n)$. If you can show that $(1,2)$ and $(1,2,\dots,n)$ are a generating set, you now have a proof that $\sigma$ and $\tau$ are also a generating set.
Parcly Taxel
- 105,904
-
(+1). Of course, we already know that $S_n$ can be generated by $(12)$ and $(12\cdots n)$, see here, or Conrad's notes from above. – Dietrich Burde Dec 11 '20 at 12:04
-
Thank you so much! This is of great help, I see now how this can be proven. – Some_guy Dec 11 '20 at 14:22