Question

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

(a)${\mathrm{\mathbb{Z}}}_4$

Short answer

The order of every element in the group${\mathrm{\mathbb{Z}}}_4$ is$\left\{1,4,2,4\right\}$

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 ℤ4

The elements in${\mathrm{\mathbb{Z}}}_4$are $\left\{0,1,2,3\right\}$. The order of the group${\mathrm{\mathbb{Z}}}_4$is 4.

The order of every element is given below:

Elements	Order
0	1
1	4
2	2
3	4

Therefore, the order of every element in the group${\mathrm{\mathbb{Z}}}_4$ is $\left\{1,4,2,4\right\}$.