Question

Do Problem 10 for a rectangle. Note that now only two rotations and 2 reflections leave the rectangle unchanged. So you have a group of order 4. To which is it isomorphic, the cyclic group or the 4''s group?

Short answer

The solution of given statement is

$$\begin{gathered} l=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right) \\ P=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right), \\ -P=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right) \\ -l=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right) \end{gathered}$$

Step-by-step solution

Given information

The given figure is rectangle.

Definition of Symmetry Group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bisections from the set to itself, and whose group operation is the composition of functions

Verify the given statement

Consider the following rectangle,

Observe that A B C D is a rectangle, the line through the midpoint of P and Q the respective sides $\overline{AB}\mathrm{and}\overline{CD}$and is perpendicular to both$\overline{AD}\mathrm{and}\overline{BC}$ . Also A and B is equidistance fromP, reflection across I interchanges A and B, and similarly for C and D.

Thus, the reflection also interchanges the rectangle edges $\overline{AD}\mathrm{and}\overline{BC}$ and leaves fixed the edges $\overline{AB}$and $\overline{CD}$, so it preserves the entire rectangle.

The figure is shown below

Using the same argument to the perpendicular bisector m of $\overline{BC}$, figured as shown below:

In addition to the reflections $r_m\mathrm{and}{r}_2$, it can be concluded that the rigid motion $r°r_m$ interchanges A and C and interchanges B and D . This is rotation by 180 degrees around the intersection of l and m, the center of the rectangle.

Also, there is one way to send A to D (reflection about l), one way to send A to D (reflection about m ), and one way to send A to C (the 180 degrees rotation discussed in the above).

The symmetries of a rectangle are the Klein four groups. A presentation for the group is

The multiplication table is shown below:

$\begin{array}{ccccc}& 1& a& b& c\\ 1& 1& a& b& c\\ a& a& 1& c& b\\ b& b& c& 1& a\\ c& c& b& a& 1\end{array}$

A matrix representation is the four 2x2 matrices are given below:

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],\mathrm{and}\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Therefore, the symmetries of a rectangle are isomorphic to a group of order 4