Chapter 3, Problem 1E

For each group in the following list, find the order of the group and the order of each element in the group. What relation do you see between the orders of the elements of a group and the order of the group?
$Z_{12}$ , U(10), U(12), U(20), $D_4$

To determine

To calculate : The relationship between the group's element commands and the group's commands.

  $\begin{array}{l}U\left(20\right)\hfill \\ D4\hfill \\ U\left(12\right)\hfill \\ Z12 \hfill \\ U\left(10\right) \hfill \end{array}$

Here we have to make it understand and not to verify.

Answer to Problem 1E

Clearly,it can be said that commands of the elements of the group are no larger than the order of the given group.

Explanation of Solution

Given information : List of group $Z12,U\left(10\right),U\left(12\right),U\left(20\right)$ and $D4$ is given in the question.

Calculation : Element Orders:

  $\begin{array}{l}U\left(20\right):\left|1\right|:\infty ,\left|3\right|:20,\left|7\right|:20,\left|9\right|:20,\left|11\right|:20,\left|13\right|:20,\left|17\right|:20,\left|19\right|:20\hfill \\ D4:0,4,2,4,2,0,2,0\hfill \\ U\left(12\right):\left|1\right|:\infty ,\left|5\right|:12,\left|7\right|:12,\left|11\right|:12\hfill \\ Z12:0,12,6,4,3,12,2,12,6,12,6,12\hfill \\ U\left(10\right):\left|1\right|:\infty ,\left|3\right|:10,\left|7\right|:10,\left|9\right|:10\hfill \end{array}$