Question

Find the order of each element in the given group:

(b)${\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_2$

Short answer

The order of each element of ${\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_2$is calculated below.

$$\begin{gathered} \left(0,0,0\right)=1,\left(0,0,1\right)=2,\left(0,1,0\right)=3,\left(0,1,1\right)=6,\left(0,2,0\right)=3,\left(0,2,1\right)=6 \\ \left(1,0,0\right)=3\left(1,0,1\right)=6,\left(1,1,0\right)=3,\left(1,2,0\right)=3,\left(1,2,1\right)=6,\left(2,2,0\right)=3, \\ \left(2,2,1\right)=6,\left(2,1,0\right)=3,\left(2,0,1\right)=6,\left(2,1,1\right)=6,\left(2,0,0\right)=3,\left(1,1,1\right)=6 \end{gathered}$$

Step-by-step solution

Method to find Order

Order of$\left(a,b\right)$ in$G\times G\text{'}$ is equal to LCM (order of a in G, order of b in G').

Find the order of each element in the group ℤ3 

Consider the group ${\mathrm{\mathbb{Z}}}_3$.

${\mathrm{\mathbb{Z}}}_2=\left\{0,1,2\right\}$

The order of 0 is 1 .

Since, 1 added three times and yields zero modulo 3 .

1 + 1 + 1 = 3(mod3)

= 0

Therefore, the order of 1 is 3 .

Since, 2 added three times and yields zero modulo 3 .

2 + 2 + 2 = 6(mod3)

= 0

Therefore, the order of 2 is 3 .

Find the order of each element in the group ℤ2

Consider the group ${\mathrm{\mathbb{Z}}}_2$.

${\mathrm{\mathbb{Z}}}_2=\left\{0,1\right\}$

The order of 0 is 1 .

Since, 1 added two times and yields zero modulo 2 .

1 + 1 = 2(mod2)

= 0

Therefore, the order of 1 is 2 .

Find the order of each element in the group ℤ3×ℤ3×ℤ2

The elements of${\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_2$ are as follows.

role="math" localid="1653547636437" $\left\{0,1,2\right\}\times \left\{0,1,2\right\}\times \left\{0,1\right\}=\left\{\begin{array}{l}\left(0,0,0\right),\left(0,0,1\right),\left(0,1,0\right),\left(0,1,1\right),\left(0,2,0\right),\left(0,2,1\right)\\ \begin{array}{c}\left(1,0,0\right),\left(1,0,1\right),\left(1,1,0\right),\left(1,2,0\right),\left(1,2,1\right),\left(2,2,0\right)\\ \left(2,2,1\right),\left(2,1,0\right),\left(2,0,1\right),\left(2,1,1\right),\left(2,0,0\right),\left(1,1,1\right)\end{array}\end{array}\right.\left.\begin{array}{r}\\ \begin{array}{c}\\ \end{array}\end{array}\right\}$

Find the order of (0,0,0).

LCM(1, 1, 1) = 1

Therefore, the order of (0,0,0) is 1 .

Find the order of (0,0,1) .

LCM(1, 1, 2 ) = 2

Therefore, the order of (0,0,1) is 2 .

Find the order of (0,1,0) .

LCM(1,3,1) = 3

Therefore, the order of (0,1,0) is 3 .

Find the order of (0,1,1) .

LCM(1,3,2) = 6

Therefore, the order of (0,1,1) is 6 .

Find the order of (0,2,0) .

LCM(1,3,1) = 3

Therefore, the order of (0,2,0) is 3 .

Find the order of (0,2,1) .

LCM(1,3,2) = 6

Therefore, the order of (0,2,1) is 6 .

Find the order of (1,0,0) .

LCM(3,1,1) = 3

Therefore, the order of (1,0,0) is 3 .

Find the order of (1,0,1) .

LCM(3,1,2) = 6

Therefore, the order of (1,0,1) is 6 .

Find the order of (1,1,0) .

LCM(3,3,1) = 3

Therefore, the order of (1,1,0) is 3 .

Find the order of (1,2,0) .

LCM(3,3,1) = 3

Therefore, the order of (1,2,0) is 3 .

Find the order of (1,2,1) .

LCM(3,3,2) = 6

Therefore, the order of (1,2,1) is 6 .

Find the order of (2,2,0) .

LCM(3,3,1) = 3

Therefore, the order of (2,2,0) is 3 .

Find the order of (2,2,1) .

LCM(3,3,2) = 6

Therefore, the order of (2,2,1) is 6 .

Find the order of (2,1,0) .

LCM(3,3,1) = 3

Therefore, the order of (2,1,0) is 3 .

Find the order of (2,0,1) .

LCM(3,1,2) = 6

Therefore, the order of (2,0,1) is 6 .

Find the order of (2,1,1) .

LCM(3,3,2) = 6

Therefore, the order of (2,1,1) is 6 .

Find the order of (2,0,0) .

LCM(3,1,1) = 3

Therefore, the order of (2,0,0) is 3 .

Find the order of (1,1,1) .

LCM(3,3,2) = 6

Therefore, the order of (1,1,1) is 6 .