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

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 $S_p$ is the symmetric group of order $p!$. We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length $p$ generate $S_p$.

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