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}
- Explain why the order of \(G\) is 6 .
- 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 \(G\).