Here is the symmetric group of order . We must prove that an arbitrary transposition (that is, a cycle of length 2) and cycle of length generate .Show that if is prime, where is any transposition and is any -cycle.

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!