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]
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\includegraphics[alt={},max width=\textwidth]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-4_440_394_440_248}
\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]
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\includegraphics[alt={},max width=\textwidth]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-4_321_1447_1628_242}
\end{figure}
- 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 | |
- 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.
- Explain why there can be no subgroup of \(( \mathrm { G } , \circ )\) of order 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. - 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 .