Verify group axioms

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 \(G\).

7

1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
   (a) Prove that the set of real numbers, together with the operation \(*\), forms a group.
   (b) State, with a reason, whether the group is commutative.
   (c) Prove that there are no elements of order 2.
2. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.

1 Determine whether the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 1 } = \frac { z + 4 } { 2 } \quad \text { and } \quad \frac { x + 3 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 5 } { 4 }$$ intersect or are skew.
\(2 \quad H\) denotes the set of numbers of the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.

1. Show that the product of any two members of \(H\) is a member of \(H\). It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
2. State the identity element of the group.
3. Find the inverse of \(a + b \sqrt { 5 }\).
4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse.

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 .

2 It is given that the set of complex numbers of the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) for \(- \pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
2. Prove the closure property for the group.
3. \(Z\) denotes the element \(\mathrm { e } ^ { \mathrm { i } \gamma }\), where \(\frac { 1 } { 2 } \pi < \gamma < \pi\). Express \(Z ^ { 2 }\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta < 0\).

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 \(\left( S , { } _ { 18 } \right)\).

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 \(G\).

4 Let \(S\) be the set \(\{ 16,36,56,76,96 \}\) and \(\times _ { H }\) the operation of multiplication modulo 100 .

1. Given that \(a\) and \(b\) are odd positive integers, show that \(( 10 a + 6 ) ( 10 b + 6 )\) can also be written in the form \(10 n + 6\) for some odd positive integer \(n\).
2. Construct the Cayley table for \(\left( S , \times _ { H } \right)\)
3. Show that \(\left( S , \times _ { H } \right)\) is a group.
   [0pt] [You may use the result that \(\times _ { H }\) is associative on \(S\).]
4. Write down all generators of \(\left( S , \times _ { H } \right)\).

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.
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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.

8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.

6

1. Given that \(\sqrt { 7 }\) is an irrational number, prove that \(a ^ { 2 } - 7 b ^ { 2 } \neq 0\) for all \(a , b \in \mathbb { Q }\) where \(a\) and \(b\) are not both 0 .
2. A set \(G\) is defined by \(G = \{ a + b \sqrt { 7 } : a , b \in \mathbb { Q } , a\) and \(b\) not both \(0 \}\). Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.)
3. A subset \(H\) of \(G\) is defined by \(H = \{ 1 + c \sqrt { 7 } : c \in \mathbb { Q } \}\). Determine whether or not \(H\) is a subgroup of ( \(G , \times\) ).
4. Using \(( G , \times )\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false.

4 A binary operation, ○, is defined on a set of numbers, \(A\), in the following way.
\(a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }\), for \(a , b \in A\),
where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are constants (which are not necessarily in \(A\) ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and \(A\).

• \(A = \mathbb { R }\)
• \(0 \circ 0 = 2\)
• An identity element, \(e\), exists for ∘ in \(A\)
  1. Show that \(a \circ b = a + b + 2\).
  2. State the value of \(e\).
  3. Explain whether ○ is commutative over \(A\).
  4. Determine whether or not ( \(A , \circ\) ) is a group.
  5. Explain whether your answer to part (d) would change in each of the following cases, giving details of any change.
     1. \(A = \mathbb { Z }\)
     2. \(A = \{ 2 m : m \in \mathbb { Z } \}\)
     3. \(\quad A = \{ n : n \in \mathbb { Z } , n \geqslant - 2 \}\)

4 The set \(G\) is given by \(G = \{ \mathbf { M } : \mathbf { M }\) is a real \(2 \times 2\) matrix and det \(\mathbf { M } = 1 \}\).

1. Show that \(G\) forms a group under matrix multiplication, × . You may assume that matrix multiplication is associative.
2. The matrix \(\mathbf { A } _ { n }\) is defined by \(\mathbf { A } _ { n } = \left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\) for any integer \(n\). The set \(S\) is defined by \(\mathrm { S } = \left\{ \mathrm { A } _ { \mathrm { n } } : \mathrm { n } \in \mathbb { Z } , \mathrm { n } \geqslant 0 \right\}\).
   1. Determine whether \(S\) is closed under × .
   2. Determine whether \(S\) is a subgroup of ( \(G , \times\) ).
3. 1. Find a subgroup of ( \(G , \times\) ) of order 2 .
   2. By considering the inverse of the non-identity element in any such subgroup, or otherwise, show that this is the only subgroup of ( \(G , \times\) ) of order 2. The set of all real \(2 \times 2\) matrices is denoted by \(H\).
4. With the help of an example, explain why ( \(H , \times\) ) is not 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 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 group \(\mathrm { S } _ { 4 }\) is the set of all possible permutations that can be performed on the four numbers 1, 2, 3 and 4, under the operation of composition.

For the group \(\mathrm { S } _ { 4 }\)

1. write down the identity element,
2. write down the inverse of the element \(a\), where $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array} \right)$$
3. demonstrate that the operation of composition is associative using the following elements $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array} \right) \quad b = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 2 & 4 & 3 & 1 \end{array} \right) \quad \text { and } c = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \end{array} \right)$$
4. Explain why it is possible for the group \(\mathrm { S } _ { 4 }\) to have a subgroup of order 4 You do not need to find such a subgroup.

4

1. The binary operation is defined on \(\mathbb { Z }\) by \(a\) b \(b = a + b - a b\) for all \(a , b \in \mathbb { Z }\). Prove that is associative on \(\mathbb { Z }\). The operation ∘ is defined on the set \(A = \{ 0,2,3,4,5,6 \}\) by \(a \circ b = a + b - a b ( \bmod 7 )\) for all \(a , b \in A\).
2. Complete the Cayley table for \(\left( A , { } ^ { \circ } \right)\) given in the Printed Answer Booklet.
3. Prove that \(( A , \circ )\) is a group. You may assume that the operation is associative.
4. List all the subgroups of \(( A , \circ )\).

4 The set \(L\) consists of all points \(( x , y )\) in the cartesian plane, with \(x \neq 0\). The operation ◇ is defined by \(( a , b ) \diamond ( c , d ) = ( a c , b + a d )\) for \(( a , b ) , ( c , d ) \in L\).

1. 1. Show that \(L\) is closed under ◇.
   2. Prove that \(\diamond\) is associative on \(L\).
   3. Find the identity element of \(L\) under ◇ .
   4. Find the inverse element of \(( a , b )\) under ◇.
2. Find a subgroup of \(( L , \diamond )\) of order 2.

8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\).
   [0pt] [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.

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 \(n \in \mathbb { Z }\). 1 (a) A group \(G\) of order 6 has the combination table shown below. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
3. 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 \(Q\). 7 Three planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in a line \(l\); planes \(\Pi _ { 2 }\) and \(\Pi _ { 3 }\) intersect in a line \(m\).
4. Show that \(l\) and \(m\) are in the same direction.
5. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
6. By considering the cartesian equations of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\), or otherwise, determine whether or not the three planes have a common line of intersection. 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 \(( 1,0 )\).
7. Prove that \(*\) is associative.
8. Find all the elements which commute with \(( 1,1 )\).
9. 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 \(n\).
10. Find all self-inverse elements.
11. Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.

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

1. Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
   5
2. In the multiplicative group of integers modulo 13, the group \(G\) is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5
3. 1. Explain why \(G\) is an abelian group.
      5
4. (ii) Find the order of \(G\).
   5
5. State the identity element of \(G\) and prove it is an identity element. Fully justify your answer.
   5
6. Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13