8.03a Binary operations: and their properties on given sets

4 A binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.

1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
   (B) Show that \(S\), with the operation \(*\), is a group.
4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.

   \(\circ\)	0	1	2	4	5	6
   0	0	1	2	4	5	6
   1	1	4	0	6	2	5
   2	2	0	5	1	6	4
   4	4	6	1	5	0	2
   5	5	2	6	0	4	1
   6	6	5	4	2	1	0

6. Find the order of each element of \(G\).
7. List all the subgroups of \(G\).

4

1. The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
   1. Demonstrate associativity for the elements \(3,9,11\) in that order. Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
   2. Write down the order of each element.
   3. Write down all subgroups of \(P\).
   4. Show that the group in part (i) is cyclic.
2. Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
3. Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & - 1 \end{array} \right) .$$ The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
4. The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\). By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\). Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.

1. Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
2. Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
3. Write down the order of each element of the group.
4. Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
5. 1. Give an example of a non-cyclic group of order 6 .
   2. Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).

7 The diagram below shows an equilateral triangle \(A B C\). The three lines of reflection symmetry of \(A B C\) (the lines \(a , b\) and \(c\) ) are shown as broken lines. The point of intersection of these three lines, \(O\), is the centre of rotational symmetry of the triangle. \includegraphics[max width=\textwidth, alt={}, center]{06496165-0b83-4050-ae26-fa5a0614bd46-4_533_538_884_246} The group \(D _ { 3 }\) is defined as the set of symmetries of \(A B C\) under the composition of the following transformations. \(i\) : the identity transformation \(a\) : reflection in line \(a\) \(b\) : reflection in line \(b\) \(c\) : reflection in line \(c\) \(p\) : an anticlockwise rotation about \(O\) through \(120 ^ { \circ }\) \(q\) : a clockwise rotation about \(O\) through \(120 ^ { \circ }\) Note that the lines \(a , b\) and \(c\) are unaffected by the transformations and remain fixed.

1. On the diagrams provided in the Printed Answer Booklet, show each of the six elements of \(D _ { 3 }\) obtained when the above transformations are applied to triangle \(A B C\).
2. Complete the Cayley table given in the Printed Answer Booklet.
3. List all the proper subgroups of \(D _ { 3 }\).
4. State, with justification, whether \(D _ { 3 }\) is
   1. cyclic,
   2. abelian.
5. The group \(H\), also of order 6, is the set of rotational symmetries of the regular hexagon. Describe two structural differences between \(D _ { 3 }\) and \(H\). \section*{END OF QUESTION PAPER}

7 The group \(G\), of order 12, consists of the set \(\{ 1,2,4,5,8,10,13,16,17,19,20 , x \}\) under the operation of multiplication modulo 21 . The identity of \(G\) is the element 1 . The element \(x\) is an integer, \(0 < x < 21\), distinct from the other elements in the set. An incomplete copy of the Cayley table for \(G\) is shown below:

1. State, with justification, the value of \(x\).
2. In the table given in the Printed Answer Booklet, list the order of each of the non-identity elements of \(G\).
3. 1. Write down all the subgroups of \(G\) of order 3 .
   2. Write down all the subgroups of \(G\) of order 6 .
4. Determine all the subgroups of \(G\) of order 4, and prove that there are no other subgroups of order 4.
5. State, with a reason, whether \(G\) is a cyclic group.

4

1. For the set \(S = \{ 2,4,6,8,10,12 \}\), under the operation \(\times _ { 14 }\) of multiplication modulo 14, complete the Cayley table given in the Printed Answer Booklet.
2. Show that ( \(S , \times _ { 14 }\) ) forms a group, \(G\). (You may assume that \(\times _ { 14 }\) is associative.)
3. 1. Write down all the proper subgroups of \(G\).
   2. Given that \(G\) is cyclic, write down all the possible generators of \(G\).

6 The binary operation ◇ is defined on the set \(\mathbb { C }\) of complex numbers by \(( a + i b ) \diamond ( c + i d ) = a c + i ( b + a d )\) where \(a , b , c\) and \(d\) are real numbers.

1. Is \(\mathbb { C }\) closed under △ ? Justify your answer.
2. Prove that ◇ is associative on \(\mathbb { C }\).
3. Determine the identity element of \(\mathbb { C }\) under \(\diamond\).
4. Determine the largest subset S of \(\mathbb { C }\) such that \(( \mathrm { S } , \diamond )\) is a group.

5 The binary operation * is defined as $$a * b = a + b + 4 ( \bmod 6 )$$ where \(a , b \in \mathbb { Z }\). 5

1. Show that the set \(\{ 0,1,2,3,4,5 \}\) forms a group \(G\) under *.
   5
2. Find the proper subgroups of the group \(G\) in part (a).
   5
3. Determine whether or not the group \(G\) in part (a) is isomorphic to the group \(K = \left( \langle 3 \rangle , \times _ { 14 } \right)\) [0pt] [3 marks]

1. The set \(\{ e , p , q , r , s \}\) forms a group, \(A\), under the operation *

Given that \(e\) is the identity element and that $$p ^ { * } p = s \quad s ^ { * } s = r \quad p ^ { * } p ^ { * } p = q$$

1. show that
   1. \(p ^ { * } q = r\)
   2. \(s ^ { * } p = q\)
2. Hence complete the Cayley table below.

   *	\(e\)	\(\boldsymbol { p }\)	\(\boldsymbol { q }\)	\(r\)	\(s\)
   \(e\)
   \(\boldsymbol { p }\)
   \(\boldsymbol { q }\)
   \(\boldsymbol { r }\)
   \(S\)

   A spare table can be found on page 11 if you need to rewrite your Cayley table.
3. Use your table to find \(p ^ { * } q ^ { * } r ^ { * } s\) A student states that there is a subgroup of \(A\) of order 3
4. Comment on the validity of this statement, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{989d779e-c40a-4658-ad98-17a37ab1d9e1-11_2464_74_304_36}
   Only use this grid if you need to rewrite the Cayley table.

   *	\(e\)	\(\boldsymbol { p }\)	\(\boldsymbol { q }\)	\(r\)	\(s\)
   \(e\)
   \(\boldsymbol { p }\)
   \(\boldsymbol { q }\)
   \(\boldsymbol { r }\)
   \(S\)

1. (i) Let \(G\) be a group of order 5291848

Without performing any division, use proof by contradiction to show that \(G\) cannot have a subgroup of order 11
(ii) (a) Complete the following Cayley table for the set \(X = \{ 2,4,8,14,16,22,26,28 \}\) with the operation of multiplication modulo 30 A copy of this table is given on page 11 if you need to rewrite your Cayley table.
(b) Hence determine whether the set \(X\) with the operation of multiplication modulo 30 forms a group.
[0pt] [You may assume multiplication modulo \(n\) is an associative operation.] Only use this grid if you need to rewrite your Cayley table. (Total for Question 3 is 9 marks)

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. 1. The table below is a Cayley table for the group \(G\) with operation ∘

1. State which element is the identity of the group.
2. Determine the inverse of the element ( \(b \circ c\) )
3. Give a reason why the set \(\{ a , b , e , f \}\) cannot be a subgroup of \(G\). You must justify your answer.
4. Show that the set \(\{ b , d , f \}\) is a subgroup of \(G\).
   (ii) Given that \(H\) is a group with an element \(x\) of order 3 and an element \(y\) of order 6 satisfying $$y x = x y ^ { 5 }$$ show that \(y ^ { 3 } x y ^ { 3 } x ^ { 2 }\) is the identity element. \includegraphics[max width=\textwidth, alt={}, center]{7d269bf1-f481-46bd-b9d3-fea211b186cf-02_2270_54_309_1980}

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. 1. A binary operation * is defined on positive real numbers by

$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7

1. Show that \(G\) is cyclic. The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
2. List all the subgroups of \(H\).
3. Describe an isomorphism between \(G\) and \(H\).

1. A binary operation ★ on the set of non-negative integers, \(\mathbb { Z } _ { 0 } ^ { + }\), is defined by

$$m \star n = | m - n | \quad m , n \in \mathbb { Z } _ { 0 } ^ { + }$$

1. Explain why \(\mathbb { Z } _ { 0 } ^ { + }\)is closed under the operation
2. Show that 0 is an identity for \(\left( \mathbb { Z } _ { 0 } ^ { + } , \star \right)\)
3. Show that all elements of \(\mathbb { Z } _ { 0 } ^ { + }\)have an inverse under ★
4. Determine if \(\mathbb { Z } _ { 0 } ^ { + }\)forms a group under ★, giving clear justification for your answer.

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.

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 \(G\).

4

1. State the definition of a bipartite graph. 4
2. A jazz quintet has five musical instruments: bassoon, clarinet, flute, oboe and violin. Jay, Kay, Lee, Mel and Nish are musicians and each plays a musical instrument in the jazz quintet. Jay knows how to play the bassoon and the clarinet.
   Kay knows how to play the bassoon, the oboe and the violin.
   Lee knows how to play the clarinet and the flute.
   Mel knows how to play the clarinet, the oboe and the violin.
   Nish knows how to play the flute, the oboe and the violin. 4 (b) (i) Draw a graph to show which musicians know how to play which instruments. 4 (b) (ii) Nish arrives late to a jazz quintet rehearsal. Each of the other four musicians is already playing an instrument: \begin{displayquote} Jay is playing the clarinet
   Kay is playing the oboe
   Lee is playing the flute
   Mel is playing the violin. \end{displayquote} Explain how the graph in part (b)(i) shows that there is no instrument available that Nish knows how to play. 4 (b) (iii) When Nish arrives the rehearsal stops. When they restart the rehearsal, Nish is playing the flute. Draw all possible subgraphs of the graph in part (b)(i) that show how Jay, Kay, Lee and Mel can each be assigned a unique musical instrument they know how to play.
   [0pt] [2 marks]

5

1. Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
   \end{figure} 5
2. The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for \(S\) under the commutative binary operation • \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2}

   •	\(a\)	\(b\)	\(c\)	\(d\)
   \(a\)	\(b\)	\(a\)	\(a\)	\(c\)
   \(b\)		\(c\)		\(c\)
   \(c\)		\(d\)	\(d\)
   \(d\)			\(d\)	\(d\)

   \end{table} 5 (b) (i) Complete the Cayley table in Figure 2. 5 (b) (ii) Determine whether the binary operation • is associative when acting on the elements of \(S\). Fully justify your answer.

8 The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows a Cayley table for \(S\) under the commutative binary operation \begin{table}[h] \end{table} 8

1. 1. Prove that there exists an identity element for \(S\) under the binary operation
      [0pt] [2 marks]
      8
      1. (ii) State the inverse of \(b\) under the binary operation
         8
   2. Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3}

      \(\times _ { 4 }\)	0	1	2	3
      0	0	0	0	0
      1	0	1	2	3
      2	0	2	0	2
      3	0	3	2	1

      \end{table} Mali says that, by substituting suitable distinct values for \(a , b , c\) and \(d\), the Cayley table in Figure 2 could represent multiplication modulo 4 Use your answers to part (a) to show that Mali's statement is incorrect. \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-20_2491_1736_219_139}

6 The set \(S\) is given by \(S = \{ \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } \}\) where \(\mathbf { A } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right]\) \(\mathbf { B } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\) \(\mathbf { C } = \left[ \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right]\) \(\mathbf { D } = \left[ \begin{array} { l l } 0 & 0 \\ 0 & 0 \end{array} \right]\) 6

1. Complete the Cayley table for \(S\) under matrix multiplication.

   	A	B	C	D
   A	A		D
   B		B
   C			C
   D				D

   6
2. Using the Cayley table above, explain why \(\mathbf { B }\) is the identity element of \(S\) under matrix multiplication.
   [0pt] [1 mark] 6
3. Sam states that the Cayley table in part (a) shows that matrix multiplication is commutative. Comment on the validity of Sam's statement.

5

1. The set \(S\) is defined as \(S = \{ 0,1,2,3,4,5 \}\) 5
   1. (i) State the identity element of \(S\) under the operation multiplication modulo 6 5
   2. (ii) An element \(g\) of a set is said to be self-inverse under a binary operation * if $$g * g = e$$ where \(e\) is the identity element of the set. Find all the self-inverse elements in \(S\) under the operation multiplication modulo 6
      5
   3. \(\quad\) The set \(T\) is defined as $$T = \{ a , b , c \}$$ Figure 1 shows a partially completed Cayley table for \(T\) under the commutative binary operation - \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1}

      -	\(a\)	\(b\)	c
      \(a\)	\(a\)	c	b
      \(b\)		\(b\)	\(a\)
      c			c

      \end{table} 5
   4. 1. Complete the Cayley table in Figure 1 5
   5. (ii) Prove that is not associative when acting on the elements of \(T\)