       Next: Solution of problem 2 Up: Solution of the exercises Previous: Solution of the exercises

## Solution of problem 1

Solution 1A. Symmetry of the square. For the problems, see p. .

(i)
The symmetry operations of the square are:
1. the mapping 1 1, 2 2, 3 3, and 4 4;
2. the mapping 1 3, 2 4, 3 1, and 4 2;
3. the mapping 1 2, 2 3, 3 4, and 4 1;
4. the mapping 1 4, 2 1, 3 2, and 4 3;
5. the mapping 1 2, 2 1, 3 4, and 4 3;
6. the mapping 1 4, 2 3, 3 2, and 4 1;
7. the mapping 1 3, 3 1, which maps the points 2 and 4 onto themselves (leaves them invariant);
8. the mapping 2 4, 4 2, which maps the points 1 and 3 onto themselves (leaves them invariant).

(ii)
The symmetry operations are, respectively:

(a) the identity operation 1, (b) the two-fold rotation 2,

(c) the four-fold rotation 4 = (anti-clockwise),

(d) the four-fold rotation (clockwise),

(e) the reflection in the line ,

(f) the reflection in the line ,

(g) the reflection in the line ,

(h) and the reflection in the line .

(iii)
The orders of these symmetry operations are, respectively:

1, 2, 4, 4, 2, 2, 2, and 2.

(iv)
There are altogether 8 symmetry operations.

Solution 1B. Symmetry of the square. For the problems, see p. .

(v)
The determination of the matrix-column pairs is particularly easy because the origin is a fixed point under all symmetry operations of the square. Therefore, for all of them w = o holds. The images of the coordinate points 1,0 and 0,1 and their coordinates are easily found visually. The matrices are: (vi)
The multiplication table of the group of the square is Remarkable properties of the multiplication table are

1. If there is a `1' in the main diagonal of the table, then the element is the unit element or has order 2 and vice versa. This is easy to see.
2. One finds that in each row and in each column each element of the group occurs exactly once. This is a property of the multiplication table of any group. It is not difficult to prove but the proof needs elementary group theory.     Next: Solution of problem 2 Up: Solution of the exercises Previous: Solution of the exercises