Question

What is the order of the group ${\mathbf{U}}_{\mathbf{5}}\mathbf{\times }{\mathbf{U}}_{\mathbf{6}}\mathbf{\times }{\mathbf{U}}_{\mathbf{7}}\mathbf{\times }{\mathbf{U}}_{\mathbf{8}}$?

Short answer

Answer:

The order of group is $U_5\times U_6\times U_7\times U_8$is 192.

Step-by-step solution

Multiplicative Group

${\mathbf{U}}_{\mathbf{n}}$is the multiplicative group of units in ${\mathbf{Z}}_{\mathbf{n}}$. If ${\mathbf{G}}_{\mathbf{1}}\mathbf{\times }{\mathbf{G}}_{\mathbf{2}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\mathbf{G}}_{\mathbf{n}}$are finite groups then order of ${\mathbf{G}}_{\mathbf{1}}\mathbf{\times }{\mathbf{G}}_{\mathbf{2}}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\mathbf{G}}_{\mathbf{n}}$is $\mathbf{\left|}{\mathbf{G}}_{\mathbf{1}}\mathbf{\right|}\mathbf{\times }\mathbf{\left|}{\mathbf{G}}_{\mathbf{2}}\mathbf{\right|}\mathbf{\times }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }\mathbf{\left|}{\mathbf{G}}_{\mathbf{n}}\mathbf{\right|}$.

Find the order of the group U5×U6×U7×U8

Find the order $U_5$of as: $U_5=\left\{1,2,3,4\right\}$

The number of elements in $U_4$is 4.

Thus, the order of $U_5$is 4.

Find the order of localid="1659491601742" $U_6$as: $U_6=\left\{1,5\right\}$

The number of elements in $U_6$is 2.

Thus, the order of $U_6$is 2.

Find the order of $U_7$as: $U_7=\left\{1,2,3,4,5,6\right\}$

The number of elements in $U_7$is 6.

Thus, the order of $U_7$is 6.

Find the order of $U_8$as: $U_8=\left\{1,3,5,7\right\}$

The number of elements in $U_8$is 4.

Thus, the order of $U_8$is 4.

Find the order of the given group as follows:

$$\begin{gathered} U_5\times U_6\times U_7\times U_8=4\times 2\times 6\times 4 \\ =192 \end{gathered}$$

Therefore, the order of group $U_5\times U_6\times U_7\times U_8$is 192.