Groups of symmetries

7 The diagram below shows an equilateral triangle \(A B C\). The three lines of reflection symmetry of \(A B C\) (the lines \(a , b\) and \(c\) ) are shown as broken lines. The point of intersection of these three lines, \(O\), is the centre of rotational symmetry of the triangle.
\includegraphics[max width=\textwidth, alt={}, center]{06496165-0b83-4050-ae26-fa5a0614bd46-4_533_538_884_246} The group \(D _ { 3 }\) is defined as the set of symmetries of \(A B C\) under the composition of the following transformations.
\(i\) : the identity transformation
\(a\) : reflection in line \(a\)
\(b\) : reflection in line \(b\)
\(c\) : reflection in line \(c\)
\(p\) : an anticlockwise rotation about \(O\) through \(120 ^ { \circ }\)
\(q\) : a clockwise rotation about \(O\) through \(120 ^ { \circ }\)
Note that the lines \(a , b\) and \(c\) are unaffected by the transformations and remain fixed.

1. On the diagrams provided in the Printed Answer Booklet, show each of the six elements of \(D _ { 3 }\) obtained when the above transformations are applied to triangle \(A B C\).
2. Complete the Cayley table given in the Printed Answer Booklet.
3. List all the proper subgroups of \(D _ { 3 }\).
4. State, with justification, whether \(D _ { 3 }\) is
   1. cyclic,
   2. abelian.
5. The group \(H\), also of order 6, is the set of rotational symmetries of the regular hexagon. Describe two structural differences between \(D _ { 3 }\) and \(H\). \section*{END OF QUESTION PAPER}

3 Fig. 3.1 shows an equilateral triangle, with vertices \(\mathrm { A } , \mathrm { B }\) and C , and the three axes of symmetry of the triangle, \(\mathrm { S } _ { \mathrm { a } } , \mathrm { S } _ { \mathrm { b } }\) and \(\mathrm { S } _ { \mathrm { c } }\). The axes of symmetry are fixed in space and all intersect at the point O . \begin{figure}[h] \end{figure} There are six distinct transformations under which the image of the triangle is indistinguishable from the triangle itself, ignoring labels.
These are denoted by \(\mathrm { I } , \mathrm { M } _ { a ^ { \prime } } \mathrm { M } _ { \mathrm { b } ^ { \prime } } , \mathrm { M } _ { \mathrm { c } ^ { \prime } } , \mathrm { R } _ { 120 }\) and \(\mathrm { R } _ { 240 }\) where

• I is the identity transformation
• \(\mathrm { M } _ { \mathrm { a } }\) is a reflection in the mirror line \(\mathrm { S } _ { \mathrm { a } }\) (and likewise for \(\mathrm { M } _ { \mathrm { b } }\) and \(\mathrm { M } _ { \mathrm { c } }\) )
• \(\mathrm { R } _ { 120 }\) is an anticlockwise rotation by \(120 ^ { \circ }\) about O (and likewise for \(\mathrm { R } _ { 240 }\) ).

Composition of transformations is denoted by ○.
Fig. 3.2 illustrates the composition of \(R _ { 120 }\) followed by \(R _ { 240 }\), denoted by \(R _ { 240 } \circ R _ { 120 }\). This shows that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 }\) is equivalent to the identity transformation, so that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 } = \mathrm { I }\). \begin{figure}[h] \end{figure}

1. Using the blank diagrams in the Printed Answer Booklet, find the single transformation which is equivalent to each of the following.
   • \(M _ { a } \circ M _ { a }\)
   • \(M _ { b } \circ M _ { a }\)
   • \(\mathrm { R } _ { 120 } \circ \mathrm { M } _ { \mathrm { a } }\)
   The set of the six transformations is denoted by G and you are given that \(( \mathrm { G } , \circ )\) is a group. The table below is a mostly empty composition table for \(\circ\). The entry given is that for \(R _ { 240 } \circ R _ { 120 }\).
   First transformation performed is
   followed by

   ◯	I	\(\mathrm { M } _ { \mathrm { a } }\)	\(\mathrm { M } _ { \mathrm { b } }\)	\(\mathrm { M } _ { \mathrm { c } }\)	\(\mathrm { R } _ { 120 }\)	\(\mathrm { R } _ { 240 }\)
   I
   \(\mathrm { M } _ { \mathrm { a } }\)
   \(\mathrm { M } _ { \mathrm { b } }\)
   \(\mathrm { M } _ { \mathrm { c } }\)
   \(\mathrm { R } _ { 120 }\)
   \(\mathrm { R } _ { 240 }\)					I

2. Complete the copy of this table in the Printed Answer Booklet. You can use some or all of the spare copies of the diagram in the Printed Answer Booklet to help.
3. Explain why there can be no subgroup of \(( \mathrm { G } , \circ )\) of order 4.
4. A student makes the following claim.
   "If all the proper non-trivial subgroups of a group are abelian then the group itself is abelian."
   Explain why the claim is incorrect, justifying your answer fully.
5. With reference to the order of elements in the groups, explain why ( \(\mathrm { G } , \circ\) ) is not isomorphic to \(\mathrm { C } _ { 6 }\), the cyclic group of order 6 .

2. \begin{figure}[h] \end{figure} Figure 1 shows an equilateral triangle \(A B C\). The lines \(x , y\) and \(z\) and their point of intersection, \(O\), are fixed in the plane. The triangle \(A B C\) is transformed about these fixed lines and the fixed point \(O\). The lines \(x , y\) and \(z\) each pass through a vertex of the triangle and the midpoint of the opposite side. The transformations \(I , X , Y , Z , R _ { 1 }\) and \(R _ { 2 }\) of the plane containing triangle \(A B C\) are defined as follows:

• I: Do nothing
• \(X\) : Reflect in the line \(x\)
• \(Y\) : Reflect in the line \(y\)
• \(Z\) : Reflect in the line \(z\)
• \(R _ { 1 }\) : Rotate \(120 ^ { \circ }\) anticlockwise about \(O\)
• \(R _ { 2 }\) : Rotate \(240 ^ { \circ }\) anticlockwise about \(O\)

The operation * is defined as 'followed by' on the set \(T = \left\{ I , X , Y , Z , R _ { 1 } , R _ { 2 } \right\}\).
For example, \(X { } ^ { * } Y\) means a reflection in the line \(x\) followed by a reflection in the line \(y\).

1. 1. Complete the Cayley table on page 5 Given that the associative law is satisfied,
   2. show that \(T\) is a group under the operation *
2. Show that the element \(R _ { 2 }\) has order 3
3. Explain why \(T\) is not a cyclic group.
4. Write down the elements of a subgroup of \(T\) that has order 3

   \multirow{2}{*}{}		Second transformation
   	*	I	\(X\)	\(Y\)	\(Z\)	\(R _ { 1 }\)	\(R _ { 2 }\)
   \multirow{6}{*}{First Transformation}	I
   	\(X\)		I			\(Z\)
   	\(Y\)
   	\(Z\)
   	\(R _ { 1 }\)		\(Y\)
   	\(R _ { 2 }\)

   \footnotetext{Turn over for a spare table if you need to re-write your Cayley table } \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Only use this grid if you need to re-write your Cayley table}

   \multirow{2}{*}{}		Second transformation
   	*	I	\(X\)	\(Y\)	Z	\(R _ { 1 }\)	\(R _ { 2 }\)
   \multirow{6}{*}{First Transformation}	I
   	X		I			Z
   	Y
   	Z
   	\(R _ { 1 }\)		\(Y\)
   	\(R _ { 2 }\)

   \end{table}

6. \begin{figure}[h] \end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).

1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
3. Explain why each of the following statements is false, making your reasoning clear.
   1. \(G\) has a subgroup of order 4
   2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
   \section*{Diagram 1} \section*{Spare copy of Diagram 1}
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
   Only use this diagram if you need to redraw your answer to part (d).

4 The group \(G\) consists of the symmetries of the equilateral triangle \(A B C\) under the operation of composition of transformations (which may be assumed to be associative). Three elements of \(G\) are

• \(\boldsymbol { i }\), the identity
• \(\boldsymbol { j }\), the reflection in the vertical line of symmetry of the triangle
• \(\boldsymbol { k }\), the anticlockwise rotation of \(120 ^ { \circ }\) about the centre of the triangle.

These are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}

1. Explain why the order of \(G\) is 6 .
2. Determine
   • the order of \(\boldsymbol { j }\),
   • the order of \(\boldsymbol { k }\).
   • - Express, in terms of \(\boldsymbol { j }\) and/or \(\boldsymbol { k }\), each of the remaining three elements of \(G\).
   • Draw a diagram for each of these elements.
   • Is the operation of composition of transformations on \(G\) commutative? Justify your answer.
   • List all the proper subgroups of \(G\).