Question

List the distinct conjugacy classes of the group${\mathit{S}}_{\mathbf{4}}$ .

Short answer

Conjugacy classes of $S_4$is

$\left\{\begin{array}{l}\left\{\left(1\right)\right\},\left\{\left(12\right),\left(13\right),\left(14\right),\left(23\right),\left(24\right),\left(34\right)\right\}\\ \left\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\},\\ \left\{\left(1234\right),\left(1243\right),\left(1423\right),\left(1432\right),\left(1324\right),\left(1342\right)\right\},\\ \left\{\left(12\right)\left(34\right),\left(14\right)\left(23\right),\left(13\right)\left(24\right)\right\},\end{array}\left.\begin{array}{r}\\ \\ \\ \end{array}\right\}\right.$

Step-by-step solution

Order of S4=4!=24

$S_4=\left\{\begin{array}{l}\left(1\right),\left(12\right),\left(13\right),\left(14\right),\left(23\right),\left(24\right),\left(34\right),\\ \left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right),\\ \left(1234\right),\left(1243\right),\left(1423\right),\left(1432\right),\left(1324\right),\left(1342\right),\\ \left(12\right)\left(34\right),\left(14\right)\left(23\right),\left(13\right)\left(24\right)\end{array}\left.\begin{array}{r}\\ \\ \\ \end{array}\right\}\right.$

Conjugacy class

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

The conjugacy class of an element a consists of all the elements in G that are conjugate to a

Also we know that conjugacy is equivalence class

There are 5 distinct conjugacy class of $S_4$and is given by

$\left\{\begin{array}{l}\left\{\left(1\right)\right\},\left\{\left(12\right),\left(13\right),\left(14\right),\left(23\right),\left(24\right),\left(34\right)\right\},\\ \left\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\},\\ \left\{\left(1234\right),\left(1243\right),\left(1423\right),\left(1432\right),\left(1324\right),\left(1342\right)\right\}\\ \left\{\left(12\right)\left(34\right),\left(14\right)\left(23\right),\left(13\right)\left(24\right)\right\}\end{array}\left.\begin{array}{r}\\ \\ \\ \end{array}\right\}\right.$