Abel's theorem in problems and solutions - School of Mathematics

12 Chapter 1
for example, the composition We can imagine that the axis is
sent by the transformation into a new position (i.e., in the original
position of the axis and after this, consider the transformation as
the reflection with respect to the new position of the axis (i.e., with
respect to the original axis On the other hand, it is also possible
to consider that the axes are not rigidly fixed to the figure, and that
they do not move with it; therefore in the example which we examine,
after the transformation the transformation is done as the reflection
with respect to the original axis We will consider the compositions of
two transformations in exactly this way. With this choice the reasoning
about the vertices of the figure, analogously to the arguments presented
immediately before Problem 2, is correct. It is convenient to utilize such
arguments to calculate the multiplication table.
3. Write the multiplication table for all symmetries of the equilateral
triangle.
EXAMPLE 3. Let and denote the rotations of a square by 0°,
180°, 90° and 270° in the direction shown by the arrow (Figure 3).
FIGURE 3 FIGURE 4
4. Write the multiplication table for the rotations of the square.
EXAMPLE 4. Let and denote the reflections of the square with
respect to the axes shown in Figure 4.
5. Write the multiplication table for all symmetries of the square.
EXAMPLE 5. Let ABCD be a rhombus, which is not a square.
6. Find all symmetries of the rhombus and write their multiplication
table.
EXAMPLE 6. Let ABCD be a rectangle, which is not a square.