Showing any transposition and p-cycle generate S_p

[jackmell]

I was hoping someone could help me understand the following proof from:

http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/genset.pdf

Corollary 2.10. For a prime number $p$, $S_p$ is generated by any transposition and any p-cycle. Proof. Any p-cycle can be written as (12...p) by relabeling the objects being permuted (that means by applying an overall conjugation on $S_p$), so to show any transposition and any p-cycle generate $S_p$ it suffices to show any transposition and the standard p-cycle (12...p) generate $S_p$

The problem I'm having is the line ``Any p-cycle can be written as $(1\;2\;\cdots\;p)$ by relabeling the objects being permuted by applying an overall conjugation on $S_p$''.

An example helps me to understand:

Suppose I have $R=(3\;5),Q=(4\;1\;2\;5\;3)\in S_5$. Now, I take this to mean if I conjugate the entire group, $\sigma S_5 \sigma^{-1}$ such that $\sigma Q \sigma^{-1}=(1\;2\;3\;4\;5)$, then I've effectively ``relabeled'' $Q$ with $(1\;2\;3\;4\;5)$. That's easy to accomplish since in general for $\rho=(a\;b\;c\;d\;e)$, $\sigma\rho\sigma^{-1}=(\sigma(a)\;\sigma(b)\;\sigma(c)\;\sigma(d)\;\sigma(e))$. Then let $\sigma=(1\;2\;3\;5\;4)$ and then we have $(1\;2\;3\;5\;4)(4\;1\;2\;5\;3)(4\;5\;3\;2\;1)=(1\;2\;3\;4\;5)$. Therefore I assume this all means:

$\big<(3\;5),(4\;1\;2\;5\;3)\big>=\big<(h\;k),(1\;2\;3\;4\;5)\big>$

since conjugation is an automorphism (it's a bijection) and where $(h\;k)$ is the conjugation of $(3\;5)$ and just accepting for the moment if I can get it to that form, it generates the group via the second part of the theorem which I will accept for now.

Am I interpreting this correctly?

Thanks for reading,
Jack

 

[andrewkirk]

Yes.

The second part of the theorem is proven as Theorem 2.8 of what you linked.

Given that, you just need to write out the formulas by which an arbitrary element g of the group can be expressed in terms of a transposition T and a p-cycle C by first using Theorem 2.8 to express the conjugate of g in terms of another transposition (why not choose (1 2) WLOG) and the standard p-cycle.