Find two Sylow $3$-subgroups of $S_4$, $P$ and $Q$ and show that they are conjugate by finding an element $g\in S_4$ such that $P=gQg^{−1}$.

Question

The task is:

Find two Sylow $3$-subgroups of $S_4$, $P$ and $Q$ and show that they are conjugate by finding an element $g\in S_4$ such that $P=gQg^{−1}$.

I realize that they must be conjugate by Sylow theorems, but I'm having trouble finding an element such that $\langle (1,2,3) \rangle$ is conjugate to $\langle (2,3,4) \rangle$

Do I just brute force all elements? There must be a better way of finding this.

Also, is there a way of finding all Sylow $p$-subgroups of $S_n$ efficiently?

I'd appreciate the help since these kinds of questions seem to keep coming up on exams, but I feel like I would blunder it if I went the brute-force way.

Answer 1

Since $$(14)(123)(14)^{-1}=(1)(234),$$ the element you're looking for is $(14)$.

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Here is a lemma that will help you answer similar questions.