Question

Find the order of each element in the given group:

(c)$D_4\times {\mathrm{\mathbb{Z}}}_2$

Short answer

The order of each element of $D_4\times {\mathrm{\mathbb{Z}}}_2$ is calculated below.

$$\begin{gathered} \left(e,0\right)=1,\left(e,1\right)=2,\left(x,0\right)=2,\left(x,1\right)=2,\left(y,0\right)=4,\left(y,1\right)=4, \\ \left(y^2,0\right)=2,\left(y^2,1\right)=2,\left(y^3,0\right)=4,\left(y^3,1\right)=4,\left(xy,0\right)=2, \\ \left(xy,1\right)=2,\left(x{y}^2,0\right)=2,\left(x{y}^2,1\right)=2,\left(x{y}^3,0\right)=2,\left(x{y}^3,1\right)=2 \end{gathered}$$

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 D4

Consider the group $D_4$.

$D_4=\left\{e,x,y,y^2,y^3,xy,x{y}^2,x{y}^3\right\}$

Such that

$\begin{array}{rcl}{x}^2& =& e\\ y^4& =& e\\ x{y}^3& =& yx\end{array}$

Order of e is 1 .

Since, $x^2=e$, therefore, the order of x is 2 .

Since, $y^4=e$, therefore, the order of y is 4 .

Since, ${\left(y^2\right)}^2=e$, therefore, the order of$y^2$is 2 .

Since, ${\left(y^3\right)}^4=e$, therefore, the order of$y^3$is 4 .

Since, ${\left(xy\right)}^2=xyxy$.

It is given $\left(x{y}^3=yx\right)$. Therefore,

$\begin{array}{rcl}{\left(xy\right)}^2& =& xx{y}^{3}y\\ & =& x^2{y}^4\\ & =& e\end{array}$

Therefore, the order of xy is 2 .

Since,

$\begin{array}{rcl}{\left(x{y}^2\right)}^2& =& x{y}^{2}x{y}^2\\ & =& xyyx{y}^2\end{array}$

It is given $\left(x{y}^3=yx\right)$.

$\begin{array}{rcl}xyx{y}^3{y}^2& =& xyxy\\ & =& {\left(xy\right)}^2\\ & =& e\end{array}$

Therefore, the order of$x{y}^2$ is 2 .

Since, ${\left(x{y}^3\right)}^2=x{y}^{3}x{y}^3$.

It is given $\left(x{y}^3=yx\right)$.

$\begin{array}{rcl}{\left(x{y}^3\right)}^2& =& x{y}^{3}yx\\ & =& x{y}^{4}x\\ & =& x^2\\ & =& e\end{array}$

Therefore, the order of$x{y}^3$ is 2 .

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 .

$\begin{array}{rcl}1+1& =& 2\left(\mathrm{mod}2\right)\\ & =& 0\end{array}$

Therefore, the order of 1 is 2 .

Find the order of each element in the group D4×ℤ2

The elements of $D_4\times {\mathrm{\mathbb{Z}}}_2$are as follows.

$\left\{e,x,y,y^2,y^3,xy,x{y}^2,x{y}^3\right\}\times \left\{0,1\right\}=\left\{\begin{array}{l}\left(e,0\right),\left(e,1\right),\left(x,0\right),\left(x,1\right),\left(y,0\right),\left(y,1\right)\\ \left(y^2,0\right),\left(y^2,1\right),\left(y^3,0\right),\left(y^3,1\right),\left(xy,0\right),\\ \left(xy,1\right),\left(x{y}^2,0\right),\left(x{y}^2,1\right),\left(x{y}^3,0\right),\left(x{y}^3,1\right)\end{array}\right.\left.\begin{array}{r}\\ \\ \end{array}\right\}$

Find the order of (e, 0) .

LCM(1, 1) = 1

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

Find the order of (e, 1).

LCM(1, 2 ) = 2

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

Find the order of (x, 0) .

LCM(1, 2) = 2

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

Find the order of (x, 1) .

LCM(2, 2) = 2

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

Find the order of (y, 0) .

LCM(4, 1) = 4

Therefore, the order of (y, 0) is 4 .

Find the order of (y, 1) .

LCM(4, 2) = 4

Therefore, the order of (y, 1) is 4 .

Find the order of $\left(y^2,0\right)$.

LCM(2, 1) = 2

Therefore, the order of$\left(y^2,0\right)$is 2 .

Find the order of$\left(y^2,1\right)$.

LCM(2, 2) = 2

Therefore, the order of$\left(y^2,1\right)$is 2 .

Find the order of$\left(y^3,0\right)$.

LCM(4, 1) = 4

Therefore, the order of$\left(y^3,0\right)$is 4 .

Find the order of$\left(y^3,1\right)$.

LCM(4, 2) = 4

Therefore, the order of$\left(y^3,1\right)$is 4 .

Find the order of (xy, 0) .

LCM(2, 1) = 2

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

Find the order of (xy, 1) .

LCM(2, 2) = 2

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

Find the order of $\left(x{y}^2,0\right)$.

LCM(2, 1 ) = 2

Therefore, the order of$\left(x{y}^2,0\right)$ is 2 .

Find the order of $\left(x{y}^2,1\right)$.

LCM(2, 2) = 2

Therefore, the order of$\left(x{y}^2,1\right)$ is 2 .

Find the order of$\left(x{y}^3,0\right)$ .

LCM(2, 1) = 2

Therefore, the order of$\left(x{y}^3,0\right)$ is 2 .

Find the order of$\left(x{y}^3,1\right)$ .

LCM(2, 2) = 2

Therefore, the order of$\left(x{y}^3,1\right)$ is 2 .