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

Answers

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

- the mapping
**(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. .

Answers

**(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**holds. The images of the coordinate points 1,0 and 0,1 and their coordinates are easily found visually. The matrices are:**o** **(vi)**- The multiplication table of the group of the square is

Remarkable properties of the multiplication table are

- 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. - 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.

**Copyright © 2002 International Union of
Crystallography**