I am solving the following exercise:
Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of that square is a permutation of the vertices. Which permutations of the vertices will be maintained?
My knowledge: Since we just started our second chapter which covers the topic of isometry on a plane I don't have enough knowledge to conclude if my solution is sufficient enough. I would be very glad if you could comment and if necessary correct my solution.
During our lectures we made a list of isometry types:
orientation-preserving:
- translation
- rotation
orientation-reversing:
- reflection
- glide
in addition to it we said that the set of orientation-preserving isometries is a group. So using this knowledge i came to
My Solution: In the beginning i had trouble to understand the sentence because of this 'restriction' thing. After reading the sentence a couple of times I somehow figured out what the exercise wants me to do. So I thought of all isometries that transfer a square into it self. As a result we can neglect the two isometric types translation and glide. So therefore i worked with rotations ($rt$) and reflections ($rf$). There are three rotations (4th would be identity) and four reflections. Let's consider a square with the vertices 1,2,3 and 4. By applying:
$rt_1$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{array}$
$rt_2$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2 \end{array}$
$rt_3$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3 \end{array}$
a fourth rotation would be the identity so therefore $rt_4 = id$. With the reflections we have the possibility to reflect on a perpendicular (y-axis) and horizontal (x-axis) line (considering the middle of the square would be $(0,0)$) or reflect on the two diagonals. By applying the reflections we get:
$rf_1$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 4 & 3 & 2 & 1 \end{array}$
$rf_2$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 2 & 1 & 4 & 3 \end{array}$
$rf_3$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 3 & 2 & 1 & 4 \end{array}$
$rf_4$ we obtain $ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 1 & 4 & 3 & 2 \end{array}$
the permutations we can see above are the ones which will be maintained by the group $G = \{ id, rt_1, rt_2, rt_3, rf_1, rf_2, rf_3, rf_4 \}$. As we can see $G = D_4$.
HINT: To get the isometries of a plane figure try to match edges. Fix an edge of your square. You can now place this edge over any other edge, and even make a flip. This should convince you that there are $2 \times 4 =8$ transformations of the square.