Question

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

15. $G=S_4$; K is the subgroup $\left\{\left(\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$

(a) K(1234) and K(1324) .

Short answer

The given cosets are disjoint.

Step-by-step solution

Step by Step Solution Step 1: 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$\left\{\left(\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\right\}$

.

 Step 2: The cosets are disjoint

The elements in the coset$K\left(1234\right)$are $\left(1\right)\left(1234\right)=\left(1234\right)$.$\left(12\right)\left(34\right)\left(1234\right)=\left(24\right)$,$\left(13\right)\left(24\right)\left(1234\right)=\left(1432\right)$,$\left(14\right)\left(23\right)\left(1234\right)=\left(13\right)$

The elements in the coset $K\left(1324\right)$are $\left(1\right)\left(1324\right)=\left(1324\right)$,$\left(12\right)\left(34\right)\left(1324\right)=\left(1423\right)$,$\left(13\right)\left(24\right)\left(1324\right)=\left(34\right)$,$\left(14\right)\left(23\right)\left(1324\right)=\left(34\right)$.

Here, there are no identical elements in the coset $K\left(1234\right)$and $K\left(1324\right)$

Thus, the intersection of the two cosets are empty and disjoint.

Therefore, the given cosets $K\left(1234\right)$and $K\left(1324\right)$are disjoint.