Question

List four Sylow 3-subgroups of ${\mathbf{S}}_4$

Short answer

Four sylow 3-subgroup of$S_4$ are$\left\{\left(\text{1}\right)\text{,}\left(\text{123}\right)\text{,}\left(\text{132}\right)\right\}$ ,$\left\{\left(\text{1}\right)\text{,}\left(\text{134}\right)\text{,}\left(\text{143}\right)\right\}$ ,$\left\{\left(\text{1}\right)\text{,}\left(\text{124}\right)\left(\text{142}\right)\right\}$ and $\left\{\left(\text{1}\right)\text{,}\left(\text{234}\right)\left(\text{243}\right)\right\}$.

Step-by-step solution

Step-by-Step SolutionStep 1: Sylow p-Subgroup

Let $G$be a finite group and p is a prime. If$p^k$ is the largest power of p that divides $\left|G\right|$, then a subgroup of$G$ of order${\mathrm{p}}^{\mathrm{k}}$ is called a Sylow p-subgroup.

Consider a group ${\mathrm{S}}_4$.

Find the four Sylow 3-subgroups of S4

Find the order of ${\mathrm{S}}_4$as:

$\begin{array}{c}4!=24\\ =2^3\cdot 3\end{array}$

Therefore, any subgroup of order role="math" localid="1652856528481" $3$is a sylow 3-subgroup of ${\mathrm{S}}_4$.

Now, four sylow 3-subgroup of $S_4$are $\left\{\left(\text{1}\right)\text{,}\left(\text{123}\right)\text{,}\left(\text{132}\right)\right\}$, $\left\{\left(\text{1}\right)\text{,}\left(\text{134}\right)\text{,}\left(\text{143}\right)\right\}$,$\left\{\left(\text{1}\right)\text{,}\left(\text{124}\right)\left(\text{142}\right)\right\}$,and$\left\{\left(\text{1}\right)\text{,}\left(\text{234}\right)\left(\text{243}\right)\right\}$.

Hence, the four sylow 3-subgroup of $S_4$are $\left\{\left(\text{1}\right)\text{,}\left(\text{123}\right)\text{,}\left(\text{132}\right)\right\},\left\{\left(\text{1}\right)\text{,}\left(\text{134}\right)\text{,}\left(\text{143}\right)\right\},\left\{\left(\text{1}\right)\text{,}\left(\text{124}\right)\left(\text{142}\right)\right\}$and$\left\{\left(\text{1}\right)\text{,}\left(\text{234}\right)\left(\text{243}\right)\right\}$.