2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{285b6ae9-ca8f-46b7-b4ed-a3310fe4ebe6-04_568_634_248_717}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\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 \(y\).
- Complete the Cayley table on page 5
Given that the associative law is satisfied,
- show that \(T\) is a group under the operation *
- Show that the element \(R _ { 2 }\) has order 3
- Explain why \(T\) is not a cyclic group.
- 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 | | | \(Z\) | |
| \(Y\) | | | | | | |
| \(Z\) | | | | | | |
| \(R _ { 1 }\) | | \(Y\) | | | | |
| \(R _ { 2 }\) | | | | | | |
\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 | | | | | | |
| \(R _ { 1 }\) | | \(Y\) | | | | |
| \(R _ { 2 }\) | | | | | | |