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}