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Solution of problem 1

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

Answers

(i)
The symmetry operations of the square are:
  1. the mapping 1 $\longrightarrow$ 1, 2 $\longrightarrow$ 2, 3 $\longrightarrow$ 3, and 4 $\longrightarrow$ 4;
  2. the mapping 1 $\longrightarrow$ 3, 2 $\longrightarrow$ 4, 3 $\longrightarrow$ 1, and 4 $\longrightarrow$ 2;
  3. the mapping 1 $\longrightarrow$ 2, 2 $\longrightarrow$ 3, 3 $\longrightarrow$ 4, and 4 $\longrightarrow$ 1;
  4. the mapping 1 $\longrightarrow$ 4, 2 $\longrightarrow$ 1, 3 $\longrightarrow$ 2, and 4 $\longrightarrow$ 3;
  5. the mapping 1 $\longrightarrow$ 2, 2 $\longrightarrow$ 1, 3 $\longrightarrow$ 4, and 4 $\longrightarrow$ 3;
  6. the mapping 1 $\longrightarrow$ 4, 2 $\longrightarrow$ 3, 3 $\longrightarrow$ 2, and 4 $\longrightarrow$ 1;
  7. the mapping 1 $\longrightarrow$ 3, 3 $\longrightarrow$ 1, which maps the points 2 and 4 onto themselves (leaves them invariant);
  8. the mapping 2 $\longrightarrow$ 4, 4 $\longrightarrow$ 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 = $4^+$ (anti-clockwise),

(d) the four-fold rotation $4^3=4^{-1}=4^-$ (clockwise),

(e) the reflection $m_x$ in the line $m_x$,

(f) the reflection $m_y$ in the line $m_y$,

(g) the reflection $m_+$ in the line $m_+$,

(h) and the reflection $m_-$ in the line $m_-$.

(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 $O$ 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:

$ \begin{array}{r@{\hspace{0.3em}}r@{\hspace{0.3em}}r@
{\hspace{0.3em}}r}
1 = ...
...m}\begin{array}{rr} 0&1\\ 1&0 \end{array} \hspace{-0.2em}\right).
\end{array} $

(vi)
The multiplication table of the group of the square is


\begin{displaymath}\hspace{-15mm}\begin{array}{\vert c\vert cccccccc\vert} \hlin...
...m_x&4^{-1}&2&4&1\rule[-1.3ex]{0em}{2ex}\\ \hline
\end{array}. \end{displaymath}

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.


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