Question

10. Find the order of every element in each group.

(c)$S_3$

Short answer

The order of every element in the group$S_3$ is $\left\{1,2,2,2,3,3\right\}$.

Step-by-step solution

Order of an element

The order of an element x in a group G is the smallest positive integer n such that $x^n=e$. The order of the element x is denoted by $\left|x\right|$.

Order of elements in S3

$S_3$is the symmetric group of degree 3 and the number of elements in$S_3$is 6 which are $\left\{\left(\right)\left(12\right)\left(13\right)\left(23\right)\left(123\right)\left(132\right)\right\}$.

The multiplication table of $S_3$elements is given below:

Element	()	(12)	(13)	(23)	(123)	(132)
()	()	(12)	(13)	(23)	(123)	(132)
(12)	(12)	()	(123)	(132)	(23)	(13)
(13)	(23)	(132)	()	(123)	(13)	(12)
(23)	(13)	(123)	(132)	()	(12)	(23)
(123)	(123)	(13)	(12)	(23)	(132)	()
(132)	(132)	(23)	(13)	(12)	()	(123)

The order of each element in $S_3$is given in below table:

Element	Order
()	1
(12)	2
(13)	2
(23)	2
(123)	3
(132)	3

Therefore, the order of every element in the group $S_3$ is $\left\{1,2,2,2,3,3\right\}$.