Question

List the order of each element of${\mathit{A}}_{\mathbf{4}}$

Short answer

The order of elements of$A_4$are 1, 2, and 3.

Step-by-step solution

Alternating Group

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

Finding the order of each element of  A4

Since $A_4$is the group of all even permutations.

Its element can be given as,

$\left\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right),\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\right\}$

So, the order of each element can be given as,

Element (1) has order 1.

Elements $\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)$have order 2.

Elements $\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)$have order 3.