Question

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

(b)${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$

Short answer

The order of every element in the group${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$ is $\left\{1,4,2,4,2,2,4,4\right\}$.

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×ℤ2

The elements in${\mathrm{\mathbb{Z}}}_2$are$\left\{0,1\right\}$and${\mathrm{\mathbb{Z}}}_4$are $\left\{0,1,2,3\right\}$. Then the elements in${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$will be $\left\{\left(0,0\right)\left(0,1\right)\left(1,0\right)\left(1,1\right)\left(2,0\right)\left(2,1\right)\left(3,0\right)\left(3,1\right)\right\}$.

The order of every element are given below:

Elements	Order
(0,0)	1
(0, 1)	4
(1, 0)	2
(1, 1)	4
(2,0)	2
(2, 1)	2
(3, 0)	4
(3, 1)	4

Therefore, the order of every element in the group${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$ is $\left\{1,4,2,4,2,2,4,4\right\}$.