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

Groups 11
the triangle ABC into itself. We put the transformation corresponding to
in the intersection of the row corresponding to the transformation
with the column corresponding to the transformation So, for example,
in the selected cell of Table 1 we have to put the transformation which
is obtained by first rotating the triangle by 240° and later by 120° more.
Hence is a rotation by 360°, i.e., it coincides with We obtain the
same result by the following reasoning: transformation sends vertex A
onto vertex C, and later sends C onto A. In this way the transformation
sends A onto A. In exactly the same way we obtain that B is sent
onto B, and C onto C. Hence i.e.,
2. Complete Table 1.
Any transformation of some geometrical figure into itself which maintains
the distances between all its points is called a symmetry of the given
figure. So the rotations of the equilateral triangle, considered in Example
1, are symmetries of it.
EXAMPLE 2. Besides rotations, the equilateral triangle still possesses
3 symmetries, namely, the reflections with respect to the axes and
(Figure 2). We denote these transformations by and so that
Here it is possible to imagine
the composition of two transformations in two different ways. Consider,
FIGURE 2