Question

What is the order of each group:

(c)${\mathit{A}}_{\mathbf{10}}$

Short answer

The order$A_{10}$ is 1814400.

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 of A10

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

Since for $A_{10}$,$n=10$.

Hence, the order of $A_{10}$is:

$$\begin{gathered} \frac{n!}{2}=\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2} \\ =1814400 \end{gathered}$$

The order $A_{10}$is 1814400.