8.03e Order of elements: and order of groups

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			\(Z\)
   	\(Y\)
   	\(Z\)
   	\(R _ { 1 }\)		\(Y\)
   	\(R _ { 2 }\)

   \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}

1. The operation * is defined on the set \(G = \{ 0,1,2,3 \}\) by

$$x ^ { * } y \equiv x + y - 2 x y ( \bmod 4 )$$

1. Complete the Cayley table below.

   *	0	1	2	3
   0
   1
   2
   3

2. Show that \(G\) is a group under the operation *
   (You may assume the associative law is satisfied.)
3. State the order of each element of \(G\).
4. State whether \(G\) is a cyclic group, giving a reason for your answer.

1. The operation * is defined on the set \(S = \{ 0,2,3,4,5,6 \}\) by \(x ^ { * } y = x + y = x y ( \bmod 7 )\)

1. 1. Complete the Cayley table shown above
   2. Show that \(S\) is a group under the operation *
      (You may assume the associative law is satisfied.)
2. Show that the element 4 has order 3
3. Find an element which generates the group and express each of the elements in terms of this generator.

1. The set of matrices \(G = \{ \mathbf { I } , \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E } \}\) where

$$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) \quad \mathbf { D } = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right) \quad \mathbf { E } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 1 \end{array} \right)$$ with the operation \(\otimes _ { 2 }\) of matrix multiplication with entries evaluated modulo 2 , forms a group.

1. Show that \(\mathbf { B }\) is an element of order 3 in \(G\).
2. Determine the orders of the other elements of \(G\).
3. Give a reason why \(G\) is not isomorphic to
   1. a cyclic group of order 6
   2. the group of symmetries of a regular hexagon. The group \(H\) of permutations of the numbers 1, 2 and 3 contains the following elements, denoted in two-line notation, $$\begin{array} { l l l } e = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 2 & 3 \end{array} \right) & a = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} \right) & b = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 3 & 1 & 2 \end{array} \right) \\ c = \left( \begin{array} { l l } 1 & 2 \\ 1 & 3 \\ 2 \end{array} \right) & d = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \end{array} \right) & f = \left( \begin{array} { l l } 1 & 2 \\ 3 & 2 \end{array} \right) \end{array}$$
4. Determine an isomorphism between the groups \(G\) and \(H\).

1. 1. A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication.

Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\) (ii) The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15

1. Find the order of each element of \(H\).
2. Find three subgroups of \(H\) each of order 4, and describe each of these subgroups. The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right) \\ - \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\) where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication.
3. Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.

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 \(G\).

4 M is the set of all \(2 \times 2\) matrices \(\mathrm { m } ( a , b )\) where \(a\) and \(b\) are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b \\ 0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$

1. Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
2. Determine whether the group is commutative. The set \(\mathrm { N } _ { k }\) consists of all \(2 \times 2\) matrices \(\mathrm { m } ( k , b )\) where \(k\) is a fixed positive integer and \(b\) can take any integer value.
3. Prove that \(\mathrm { N } _ { k }\) is closed under matrix multiplication if and only if \(k = 1\). Now consider the set P consisting of the matrices \(\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )\) and \(\mathrm { m } ( 1,3 )\). The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
4. Show that \(( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )\).
5. Construct the group combination table for P . The group R consists of the set \(\{ e , a , b , c \}\) combined under the operation *. The identity element is \(e\), and elements \(a , b\) and \(c\) are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
6. Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.

1. State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
2. Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
3. 1. By finding each element of \(H\), determine the order of \(H\).
   2. List all the proper subgroups of \(H\).
4. State whether each of the following statements is true or false. Give a reason for each of your answers.

6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. 1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
   2. Verify that ( \(S , \bigcirc\) ) is a group.
   3. State the order of each element of \(( S , \bigcirc )\).
2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
3. 1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
   2. List all possible generators of \(( S , \bigcirc )\).

6 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(\quad x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).

1. Let \(x \in G\). Justify the following statements:
   1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
   2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
   3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
   4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.

9

1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
2. Prove that any group in which every non-identity element is self-inverse is abelian.
3. Simon believes that if \(x\) and \(y\) are two distinct self-inverse elements of a group, then the element \(x y\) is also self-inverse. By considering the group of the six permutations of \(\left( \begin{array} { l l } 1 & 2 \end{array} \right)\), produce a counter-example to prove him wrong.
4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct self-inverse elements of \(G\). By considering the set \(H = \{ i , x , y , x y \}\), prove by contradiction that \(G\) cannot contain all self-inverse elements.

6 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

The function f is defined by \(\text{f} : x \mapsto \frac{1}{2-2x}\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\). The function g is defined by \(\text{g}(x) = \text{ff}(x)\).

1. Show that \(\text{g}(x) = \frac{1-x}{1-2x}\) and that \(\text{gg}(x) = x\). [4]

It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where \(\text{e} : x \mapsto x\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\).

1. State the orders of the elements f and g. [2]
2. The inverse of the element f is denoted by h. Find \(\text{h}(x)\). [2]
3. Construct the operation table for the elements e, f, g, h of the group \(K\). [4]

\(Q\) is a multiplicative group of order 12.

1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]

The table below shows the number of elements of \(Q\) with each possible order. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.

1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]

A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}

1. Give a reason why \(D\) is not commutative. [1]
2. Write down the orders of any possible proper subgroups of \(D\). [2]
3. List the elements of a proper subgroup which contains
   1. the element \(a\), [1]
   2. the element \(r\). [1]
4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]

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]

      \(*\)