Question

What is the order of each group:

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

Short answer

The order $A_4$is 12.

Step-by-step solution

Definition of an alternating group.

An alternating group is one whose elements are formed by taking the even permutations of a finite set S_n. An alternating set of n elements is represented by ${\mathit{A}}_{\mathbf{n}}$, where n is called the degree of the alternating set.

Order of the Alternating set

The order of the alternating set of degree n is given by $\frac{\mathbf{n}\mathbf{!}}{\mathbf{2}}$

Finding the order ofA4

The order of the alternating group of degree n is given by$\frac{n!}{2}$

Since for $A_4$,$n=4$.

Hence, the order of $A_4$can be given as:

$$\begin{gathered} \frac{n!}{2}=\frac{4\times 3\times 2\times 1}{2} \\ =12 \end{gathered}$$

The order $A_4$is 12.