Groups

7. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.

3. A two-person zero-sum game is represented by the payoff matrix for player \(A\) shown below.

1. Represent the expected payoffs to \(A\) against \(B\) 's strategies graphically and hence determine which strategy is not worth considering for player \(B\).
2. Find the best strategy for player \(A\) and the value of the game.

Let \(V\) be the subspace of \(\mathbb { R } ^ { 4 }\) spanned by $$\mathbf { v } _ { 1 } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \\ 2 \end{array} \right) , \quad \mathbf { v } _ { 2 } = \left( \begin{array} { r } - 2 \\ - 5 \\ 5 \\ 6 \end{array} \right) , \quad \mathbf { v } _ { 3 } = \left( \begin{array} { r } 0 \\ - 3 \\ 15 \\ 18 \end{array} \right) \quad \text { and } \quad \mathbf { v } _ { 4 } = \left( \begin{array} { r } 0 \\ - 2 \\ 10 \\ 8 \end{array} \right) .$$

1. Show that the dimension of \(V\) is 3 .
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2. Express \(\mathbf { v } _ { 4 }\) as a linear combination of \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\) and \(\mathbf { v } _ { 3 }\).
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3. Write down a basis for \(V\).
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   \(\_\_\_\_\) Let \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\ 2 & - 5 & - 3 & - 2 \\ 0 & 5 & 15 & 10 \\ 2 & 6 & 18 & 8 \end{array} \right)\).
4. Find the general solution of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\).
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   \(\_\_\_\_\) The set of elements of \(\mathbb { R } ^ { 4 }\) which are not solutions of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\) is denoted by \(W\).
5. State, with a reason, whether \(W\) is a vector space.
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1 The plane \(\Pi\) passes through the points with coordinates \(( 1,6,2 ) , ( 5,2,1 )\) and \(( 1,0 , - 2 )\).

1. Find a vector equation of \(\Pi\) in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
2. Find a cartesian equation of \(\Pi\).
   \(2 G\) consists of the set \(\{ 1,3,5,7 \}\) with the operation of multiplication modulo 8 .

1 The plane \(\Pi\) passes through the points with coordinates \(( 1,6,2 ) , ( 5,2,1 )\) and \(( 1,0 , - 2 )\).

1. Find a vector equation of \(\Pi\) in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
2. Find a cartesian equation of \(\Pi\).
   \(2 G\) consists of the set \(\{ 1,3,5,7 \}\) with the operation of multiplication modulo 8 .

9 The set \(C\) consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of \(C\) by \(\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }\) where \(\mathrm { a } , \mathrm { b } \in \mathrm { C }\).

1. Determine the identity element of \(C\) under ⊕.
2. For each element \(x\) in \(C\) show that it has an inverse element in \(C\).
3. Show that \(\oplus\) is associative on \(C\).
4. Explain why \(( C , \oplus )\) is not a group.
5. Find a subset, \(D\), of \(C\) such that \(( D , \oplus )\) is a group of order 3 . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
   For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
   OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.

7

1. 1. Write down the pay-off matrix for Bex. 7
2. (ii) Explain why the pay-off matrix for Bex can be written as

7

1. 1. Write down the pay-off matrix for Bex. 7
2. (ii) Explain why the pay-off matrix for Bex can be written as

5 Alex and Beth play a zero-sum game. Alex chooses a strategy P, Q or R and Beth chooses a strategy \(\mathrm { X } , \mathrm { Y }\) or Z . The table shows the number of points won by Alex for each combination of strategies. The entry for cell \(( \mathrm { P } , \mathrm { X } )\) is \(x\), where \(x\) is an integer. \begin{table}[h] \end{table} Suppose that P is a play-safe strategy.

1. 1. Determine the values of \(x\) for which the game is stable.
   2. Determine the values of \(x\) for which the game is unstable. The game can be reduced to a \(2 \times 3\) game using dominance.
2. Write down the pay-off matrix for the reduced game. When the game is unstable, Alex plays strategy P with probability \(p\).
3. Determine, as a function of \(x\), the value of \(p\) for the optimal mixed strategy for Alex. Suppose, instead, that P is not a play-safe strategy and the value of \(x\) is - 5 .
4. Show how to set up a linear programming formulation that could be used to find the optimal mixed strategy for Alex.

2 The set \(S\) is given by \(S = \{ 0,2,4,6 \}\) 2

1. Construct a Cayley table, using the grid below, for \(S\) under the binary operation addition modulo 8
   \includegraphics[max width=\textwidth, alt={}, center]{18ce34aa-e4c3-4a84-a36d-6542d2319bf5-03_561_563_607_831} 2
2. State the identity element for \(S\) under the binary operation addition modulo 8

4 The set \(S\) is defined as \(S = \{ 1,2,3,4 \}\) 4

1. Complete the Cayley Table shown below for \(S\) under the binary operation multiplication modulo 5

   \(\times _ { 5 }\)	1	2	3	4
   1
   2
   3
   4

   4
2. State the identity element for \(S\) under multiplication modulo 5 4
3. State the self-inverse elements of \(S\) under multiplication modulo 5

3 The matrix \(\mathbf { A }\) is \(\left( \begin{array} { r r r } - 1 & 2 & 4 \\ 0 & - 1 & - 25 \\ - 3 & 5 & - 1 \end{array} \right)\). Use the Cayley-Hamilton theorem to find \(\mathbf { A } ^ { - 1 }\).
\(4 T\) is the set \(\{ 1,2,3,4 \}\). A binary operation • is defined on \(T\) such that \(a \cdot a = 2\) for all \(a \in T\). It is given that ( \(T , \cdot\) ) is a group.

1. Deduce the identity element in \(T\), giving a reason for your answer.
2. Find the value of \(1 \cdot 3\), showing how the result is obtained.
3. 1. Complete a group table for ( \(T , \bullet\) ).
   2. State with a reason whether the group is abelian.

1. Let \(G\) be a group of order \(46 ^ { 46 } + 47 ^ { 47 }\)

Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of \(G\)

1. 11
2. 21

9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)

1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
   (a) The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
   (b) The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
   (c) The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 4

5 A multiplicative group \(G\) of order 9 has distinct elements \(p\) and \(q\), both of which have order 3 . The group is commutative, the identity element is \(e\), and it is given that \(q \neq p ^ { 2 }\).

1. Write down the elements of a proper subgroup of \(G\)
   (a) which does not contain \(q\),
   (b) which does not contain \(p\).
2. Find the order of each of the elements \(p q\) and \(p q ^ { 2 }\), justifying your answers.
3. State the possible order(s) of proper subgroups of \(G\).
4. Find two proper subgroups of \(G\) which are distinct from those in part (i), simplifying the elements.

7 The binary operation ∇ is defined as $$a \nabla b = a + b + a b \text { where } a , b \in \mathbb { R }$$ 7

1. Determine if \(\nabla\) is commutative on \(\mathbb { R }\) Fully justify your answer. 7
2. Prove that ∇ is associative on \(\mathbb { R }\)

8 The operation \(*\) is defined on the elements \(( x , y )\), where \(x , y \in \mathbb { R }\), by $$( a , b ) * ( c , d ) = ( a c , a d + b ) .$$ It is given that the identity element is \(( 1,0 )\).

1. Prove that \(*\) is associative.
2. Find all the elements which commute with \(( 1,1 )\).
3. It is given that the particular element \(( m , n )\) has an inverse denoted by \(( p , q )\), where $$( m , n ) * ( p , q ) = ( p , q ) * ( m , n ) = ( 1,0 ) .$$ Find \(( p , q )\) in terms of \(m\) and \(n\).
4. Find all self-inverse elements.
5. Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.

3 A non-commutative group \(G\) consists of the six elements \(\left\{ e , a , a ^ { 2 } , b , a b , b a \right\}\) where \(e\) is the identity element, \(a\) is an element of order 3 and \(b\) is an element of order 2 .
By considering the row in \(G\) 's group table in which each of the above elements is pre-multiplied by \(b\), show that \(b a ^ { 2 } = a b\).

4 The group \(G\) consists of the set \(\{ 1,3,7,9,11,13,17,19 \}\) combined under multiplication modulo 20.

1. Find the inverse of each element.
2. Show that \(G\) is not cyclic.
3. Find two isomorphic subgroups of order 4 and state an isomorphism between them.

9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3 9

1. 1. Show that \(C\) is a cyclic group.
      9
2. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\)
   9
3. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
4. 1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\)
      Fully justify your answer.
      [0pt] [2 marks]
      9
5. (ii) Determine, with a reason, whether or not \(C \cong V\)
   \(\mathbf { 9 }\) (c) The group \(G\) has order 16
   Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim.
   \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}

8 Groups \(A , B , C\) and \(D\) are defined as follows:
\(A\) : the set of numbers \(\{ 2,4,6,8 \}\) under multiplication modulo 10 ,
\(B\) : the set of numbers \(\{ 1,5,7,11 \}\) under multiplication modulo 12 ,
\(C\) : the set of numbers \(\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}\) under multiplication modulo 15,
\(D\) : the set of numbers \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) under multiplication.

1. Write down the identity element for each of groups \(A , B , C\) and \(D\).
2. Determine in each case whether the groups $$\begin{aligned} & A \text { and } B , \\ & B \text { and } C , \\ & A \text { and } C \end{aligned}$$ are isomorphic or non-isomorphic. Give sufficient reasons for your answers.
3. Prove the closure property for group \(D\).
4. Elements of the set \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 The group \(G\) consists of a set of six matrices under matrix multiplication. Two of the elements of \(G\) are \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & - 1 \\ 0 & - 1 \end{array} \right)\).

1. Determine each of the following:
   • \(\mathbf { A } ^ { 2 }\)
   • \(\mathbf { B } ^ { 2 }\)
   • Determine all the elements of \(G\).
   • State the order of each non-identity element of \(G\).
   • State, with justification, whether \(G\) is
   • abelian
   • cyclic.

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

1. Show that \(\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }\) and that \(\operatorname { gg } ( x ) = x\). 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 e : \(x \mapsto x\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\).
2. State the orders of the elements f and g .
3. The inverse of the element f is denoted by h . Find \(\mathrm { h } ( x )\).
4. Construct the operation table for the elements e, f, g, h of the group \(K\).

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

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