The Sylow subgroups of $S_4$

Given: The hypotheses and objects in the Example.

For a finite group $G$ and a prime $p$, let ${\mathrm{Syl}}_p(G)$ be the set of Sylow $p$-subgroups (def-sylow-p-subgroup). Define $n_p(G):=\mathrm{\mid }{\mathrm{Syl}}_p(G)\mathrm{\mid }\mathrm{.}$ This cardinal is defined even before existence is proved because ${\mathrm{Syl}}_p(G)$ is a subset of the finite power set of $G$; thm-sylow-first-theorem later shows it is nonzero. (The number $n_p(G)$ of Sylow $p$-subgroups).

Let $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $p\nmid m$. Then the number of Sylow $p$-subgroups satisfies $n_p(G)\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p),n_p(G)\mid m\mathrm{.}$. (Sylow III: $n_p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$ and $n_p\mid m$ when $\mathrm{\mid }G\mathrm{\mid }=p^{a}m$ with $p\nmid m$).

Let $n\in \mathbb{N}$, so that $n=\{0,1,\dots ,n-1\}$ (def-natural-numbers). The symmetric group on $n$ letters is $S_n:=\mathrm{Sym}(n)=\mathrm{Sym}(\{0,1,\dots ,n-1\}),$ the group of all bijections of $n$ under composition (def-symmetric-group), with the composition convention. (The finite symmetric group $S_n$, one-line notation, and cycle notation).

For $\sigma ,\tau \in S_n$, there is a $g\in S_n$ with $\tau =g\sigma g^{-1}$ if and only if $\sigma$ and $\tau$ have the same cycle type, including their numbers of fixed points. (Two elements of $S_n$ are conjugate if and only if they have the same cycle type).