Thread: prove S_p is generated by a p-cycle and an arbitrary transposition

1. prove S_p is generated by a p-cycle and an arbitrary transposition

Show that if $\displaystyle p$ is prime, $\displaystyle S_p=\langle\sigma,\tau\rangle$ where $\displaystyle \sigma$ is any transposition and $\displaystyle \tau$ is any $\displaystyle p$-cycle.
Here $\displaystyle S_p$ is the symmetric group of order $\displaystyle p!$. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length $\displaystyle p$ generate $\displaystyle S_p$.

I can more or less guess intuitively how these work together, but it would be an enormous task to show it. I suspect there is an easier way, using homomorphisms or some such. Any hints would be much appreciated!

2. Originally Posted by hatsoff
Here $\displaystyle S_p$ is the symmetric group of order $\displaystyle p!$. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length $\displaystyle p$ generate $\displaystyle S_p$.

I can more or less guess intuitively how these work together, but it would be an enormous task to show it. I suspect there is an easier way, using homomorphisms or some such. Any hints would be much appreciated!

I don't think there's an easy to do this: show that some power of that p-cycle times that transposition (or some already generated element) gives you all the transpositions and you're done.
Try some particular cases> p = 3,5,7...

Tonio