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}