Question

Write out the operation table for the group ${\mathit{S}}_{\mathbf{4}}\mathbf{/}\mathit{V}$.

Short answer

The operation table of$S_4/V$is given below:

$\begin{array}{l}{S}_4/V=\{\left(\begin{array}{cccc}1& 2& 3& 456\end{array}\right),\left(\begin{array}{cccc}2& 3& 1& 645\end{array}\right),\left(\begin{array}{cccc}3& 1& 2& 564\end{array}\right),\\ \left(\begin{array}{cccc}4& 5& 6& 123\end{array}\right),\left(\begin{array}{cccc}5& 6& 4& 312\end{array}\right),\left(\begin{array}{cccc}6& 4& 5& 231\end{array}\right)\}\end{array}$

Step-by-step solution

 Step 1: Referring to partof this Exercise

$V=\left\{\left(\begin{array}{l}1234\\ 1234\end{array}\right)\left(\begin{array}{l}1234\\ 2143\end{array}\right)\left(\begin{array}{l}1234\\ 3412\end{array}\right)\left(\begin{array}{l}1234\\ 4321\end{array}\right)\right\}$

S and V is a normal subgroup of$S_4$.

Finding the operation table forS4/V

As we know

$V=\left\{\left(\begin{array}{l}1234\\ 1234\end{array}\right)\left(\begin{array}{l}1234\\ 2143\end{array}\right)\left(\begin{array}{l}1234\\ 3412\end{array}\right)\left(\begin{array}{l}1234\\ 4321\end{array}\right)\right\}$

Therefore,$\left|v\right|=4$

And we also know, thatis the symmetric group and$\left|s_4\right|=4!=24$

Therefore,

$\begin{array}{c}{S}_4/V=\frac{\left|S_4\right|}{\left|V\right|}\\ =\frac{24}{4}\\ =6\end{array}$

So, the operation table of$S_4/V$must has 6 elements.

The operation table of$S_4/V$ is given below:

$\begin{array}{l}{S}_4/V=\{\left(\begin{array}{cccc}1& 2& 3& 456\end{array}\right),\left(\begin{array}{cccc}2& 3& 1& 645\end{array}\right),\left(\begin{array}{cccc}3& 1& 2& 564\end{array}\right),\\ \left(\begin{array}{cccc}4& 5& 6& 123\end{array}\right),\left(\begin{array}{cccc}5& 6& 4& 312\end{array}\right),\left(\begin{array}{cccc}6& 4& 5& 231\end{array}\right)\}\end{array}$