Question

List the elements in each group

(b) ${\mathit{A}}_{\mathbf{4}}$

Short answer

The elements in group $A_4$are:

$$\begin{gathered} A_4=\left\{\left(I\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right), \\ \left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\} \end{gathered}$$

Step-by-step solution

Alternating group

The set of all even permutations in $S_n$is called the alternating group $A_n$of degree n. The order of group $A_n$is given by$\left|A_n\right|=\frac{n!}{2}$

Elements in A4

The Group $S_4$has the elements

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

The order of group $A_4$is $\left|A_4\right|=12$

From group $S_4$, the set of all even permutation, that is $A_4$are:

$$\begin{gathered} A_4=\left\{\left(I\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right), \\ \left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\} \end{gathered}$$

Therefore, the elements in group $A_4$are:

$$\begin{gathered} A_4=\left\{\left(I\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right), \\ \left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\} \end{gathered}$$