Question

What is the order of each group:

(b)${\mathit{A}}_{\mathbf{5}}$

Short answer

The order $A_5$is 60

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 ${\mathit{S}}_{\mathbf{n}}$. 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 of A5

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

Since for $A_5$,$n=5$.

Hence, the order is:

$$\begin{gathered} \frac{n!}{2}=\frac{5\times 4\times 3\times 2\times 1}{2} \\ =60 \end{gathered}$$

The order $A_5$is 60.