8.03j Properties of groups: higher finite order or infinite order

4

1. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
2. Show that \(M\) is cyclic.
3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.

   	A	B	C	D	E	F	G	H	I	J
   A	C	J	G	H	A	B	I	F	E	D
   B	F	E	H	G	B	A	D	C	J	I
   C	G	D	I	F	C	J	E	B	A	H
   D	J	C	B	E	D	G	F	I	H	A
   E	A	B	C	D	E	F	G	H	I	J
   F	H	I	D	C	F	E	J	A	B	G
   G	I	H	E	B	G	D	A	J	C	F
   H	D	G	J	A	H	I	B	E	F	C
   I	E	F	A	J	I	H	C	D	G	B
   J	B	A	F	I	J	C	H	G	D	E

   One of these transformations is the identity transformation, some are rotations and the rest are reflections.
4. Identify which transformation is the identity, which are rotations and which are reflections.
5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
6. Find the order of each element of \(P\).
7. List all the proper subgroups of \(P\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).

1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
3. Explain why each of the following statements is false, making your reasoning clear.
   1. \(G\) has a subgroup of order 4
   2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
   \section*{Diagram 1} \section*{Spare copy of Diagram 1}
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
   Only use this diagram if you need to redraw your answer to part (d).

7 The set \(M\) contains all matrices of the form \(\mathbf { X } ^ { n }\), where \(\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)\) and \(n\) is a positive integer.

1. Show that \(M\) contains exactly 12 elements.
2. Deduce that \(M\), together with the operation of matrix multiplication, form a cyclic group \(G\).
3. Determine all the proper subgroups of \(G\). \section*{END OF QUESTION PAPER}

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}

1. Explain why the order of \(G\) is 6 .
2. 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 \(G\).

The group \(G\) has binary operation \(*\) and order \(p\), where \(p\) is a prime number.

1. Determine the number of distinct subgroups of \(G\) Fully justify your answer. [2 marks]
2. \(G\) contains an element \(g\) which has period \(p\)
   1. State the general name given to elements such as \(g\) [1 mark]
   2. State the name of a group that is isomorphic to \(G\) [1 mark]
3. \(G\) contains an element \(g^r\), where \(r < p\) Find, in terms of \(g\), \(r\) and \(p\), the inverse of \(g^r\) [2 marks]
4. In the case when \(p = 5\) and the binary operation \(*\) represents addition modulo 5, \(G\) contains the elements 0, 1, 2, 3 and 4
   1. Explain why \(G\) is closed. [1 mark]
   2. Complete the Cayley table for \((G, *)\) [1 mark]

      \(*\)