Exercise 4.5.7(Sylow $2$-subgroups of $S_4$)

Proof. By Sylow’s Theorem, the number $n_2$ of Sylow $2$-subgroups of $S_4$ satisfies

thus $n_2=1$ or $n_2=3$.

Let $H$ be the subgroup of $A_4$ given by

Then $H⊴S_4$. Consider the subgroup $K=⟨\tau ⟩=\{1,\tau \}$, where $\tau =(ab)$ is any transposition. Then $H\cap K=\{1\}$.

Since $H⊴S_4$, $\mathit{𝐻𝐾}$ is a subgroup of $S_4$, and

thus $\mathit{𝐻𝐾}$ is a $2$-Sylow of $S_4$. Moreover, since $H$ is a maximal subgroup of $\mathit{𝐻𝐾}$, and $\tau =(ab)\notin H$,

There are $6$ transpositions, but

Thus there are three Sylow $2$-subgroups of $S_4$, given by

Put $\tau =(23)$. Since $\mathit{𝜏𝐻}{\tau }^{-1}=H$,

Thus

Similarly,

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