Question

List all conjugates of sylow 3-subgroup$\left(123\right)$ in${\mathit{s}}_{\mathbf{4}}$ .

Short answer

Conjugate of sylow 3-subgroup$\left(123\right)$ in$s_4$ are

$\left\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\}$

Step-by-step solution

Order of s4=4!=24  

$s_4=\left\{\begin{array}{l}\left(1\right),\left(12\right),\left(13\right),\left(14\right),\left(23\right),\left(24\right),\left(34\right),\\ \begin{array}{c}\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right),\\ \left(12\right)\left(34\right),\left(14\right)\left(23\right),\left(13\right)\left(24\right)\end{array}\end{array}\left.\begin{array}{r}\\ \begin{array}{c}\\ \end{array}\end{array}\right\}\right.$

Conjugacy class

Let G be a group and $a,b\in G$. We say that ais conjugate tobif there exist $x\in G$such that $\mathit{b}\mathbf{=}{\mathit{x}}^{-1}\mathit{a}\mathit{x}$.

The conjugacy class of an elementaconsists of all the elements inGthat are conjugate toa.

Also we know that conjugacy is equivalence class

Conjugate of sylow 3-subgroup $\left(123\right)$in $S_4$are

$\left\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\}$