Question

Question: In Exercise 13-15, K is a subgroup of G . Determine whether the given cosets are disjoint or identical.

14.$\mathit{G}\mathbf{=}{\mathit{S}}_{\mathbf{4}}$; K is the subgroup role="math" localid="1651694385347" $\left\{\left(\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$

(a) $\mathit{K}\left(12\right)$and $\mathit{K}\left(34\right)$.

Short answer

The given cosets are identical.

Step-by-step solution

Subgroup K of G  

The elements in the group $S_4$are :

$$\begin{gathered} S_4=\left\{\left(I\right),\left(12\right),\left(13\right),\left(14\right),\left(23\right),\left(24\right),\left(34\right), \\ \left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right), \\ \left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right), \\ \left(1234\right),\left(1324\right),\left(1423\right),\left(1324\right),\left(1432\right),\left(1234\right) \\ \right\} \end{gathered}$$

Given that K is the subgroup$isthesubgroup$$\{\left(\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$

The cosets are identical

The permutation (12) can be written as follows :

$$\begin{gathered} \left(12\right)=\left(12\right)\left(34\right)\left(12\right) \\ =\left(34\right) \end{gathered}$$

Here, the permutation $\left(34\right)\in K\left(12\right)$implies that $K\left(34\right)\subset K\left(12\right)$

Thus, the intersection of the two cosets are non-empty and not disjoint.

Therefore, the given cosets$K\left(12\right)$and$K\left(34\right)$are identical.