Chapter 6.4, Problem 19E

(a)

To determine

To find: A group that is permutation isomorphic to $\left(Z_3,+\right)$ .

Answer to Problem 19E

The required group is $\left\{\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\left[\begin{array}{ccc}2& 3& 1\end{array}\right],\left[\begin{array}{ccc}3& 1& 2\end{array}\right]\right\}$ .

Explanation of Solution

Given information:

Given group is $\left(Z_3,+\right)$ .

Consider the given group $\left(Z_3,+\right)$ .

Here $Z_3=\left\{0,1,2\right\}$ .

Consider a group,

  $H=\left\{{\lambda }_a:a\in Z_3\text{s}\text{.t}\text{.}{\lambda }_a\left(x\right)=a+x\right\}$

Now, ${\lambda }_1=\left[\begin{array}{ccc}1& 2& 3\end{array}\right],{\lambda }_2=\left[\begin{array}{ccc}2& 3& 1\end{array}\right]\text{and}{\lambda }_3=\left[\begin{array}{ccc}3& 1& 2\end{array}\right]$

Thus,

  $H=\left\{\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\left[\begin{array}{ccc}2& 3& 1\end{array}\right],\left[\begin{array}{ccc}3& 1& 2\end{array}\right]\right\}$

The group that is permutation isomorphic to $\left(Z_3,+\right)$ .

Hence, the required group is $\left\{\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\left[\begin{array}{ccc}2& 3& 1\end{array}\right],\left[\begin{array}{ccc}3& 1& 2\end{array}\right]\right\}$ .

(b)

To determine

To find: A group that is permutation isomorphic to $\left(Z_5,+\right)$ .

Answer to Problem 19E

The required group is

  $\left\{\left[\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}1& 2\end{array}& 3& 4\end{array}& 5\end{array}\right],\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}2& 3\end{array}& 4\end{array}& 5& 1\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}3& 4\end{array}& 5& 1& 2\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}4& 5\end{array}& 1& 2& 3\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}5& 1\end{array}& 2& 3& 4\end{array}\right]\right\}$ .

Explanation of Solution

Given information:

Given group is $\left(Z_5,+\right)$ .

Consider the given group $\left(Z_5,+\right)$ .

Here $Z_5=\left\{0,1,2,3,4\right\}$ .

Consider a group,

  $H=\left\{{\lambda }_a:a\in Z_5\right\}$

Now,

  ${\lambda }_1=\left[\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}1& 2\end{array}& 3& 4\end{array}& 5\end{array}\right],{\lambda }_2=\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}2& 3\end{array}& 4\end{array}& 5& 1\end{array}\right],{\lambda }_3=\left[\begin{array}{cccc}\begin{array}{cc}3& 4\end{array}& 5& 1& 2\end{array}\right],{\lambda }_4=\left[\begin{array}{cccc}\begin{array}{cc}4& 5\end{array}& 1& 2& 3\end{array}\right]\text{and}{\lambda }_3=\left[\begin{array}{cccc}\begin{array}{cc}5& 1\end{array}& 2& 3& 4\end{array}\right]$

Thus,

  $H=\left\{\left[\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}1& 2\end{array}& 3& 4\end{array}& 5\end{array}\right],\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}2& 3\end{array}& 4\end{array}& 5& 1\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}3& 4\end{array}& 5& 1& 2\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}4& 5\end{array}& 1& 2& 3\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}5& 1\end{array}& 2& 3& 4\end{array}\right]\right\}$

The group that is permutation isomorphic to $\left(Z_5,+\right)$ .

Hence, the required group is

  $\left\{\left[\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}1& 2\end{array}& 3& 4\end{array}& 5\end{array}\right],\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}2& 3\end{array}& 4\end{array}& 5& 1\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}3& 4\end{array}& 5& 1& 2\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}4& 5\end{array}& 1& 2& 3\end{array}\right],\left[\begin{array}{cccc}\begin{array}{cc}5& 1\end{array}& 2& 3& 4\end{array}\right]\right\}$ .

(c)

To determine

To find: A group that is permutation isomorphic to $\left(R,+\right)$ .

Answer to Problem 19E

The required group is $H=\left\{{\lambda }_a:a\in R\right\}$ .

Explanation of Solution

Given information:

Given group is $\left(R,+\right)$ .

Consider the given group $\left(R,+\right)$ .

Here $Z_3=\left\{0,1,2\right\}$ .

Consider a group $H=\left\{{\lambda }_a:a\in R\right\}$ such that,

  ${\lambda }_a=a+x$ for all $x\in R$ .

Thus, it is an infinite group.

The group that is permutation isomorphic to $\left(R,+\right)$ .

Hence, the required group is $H=\left\{{\lambda }_a:a\in R\right\}$ .