8.03g Cyclic groups: meaning of the term

2 The set \(S = \{ a , b , c , d \}\) under the binary operation * forms a group \(G\) of order 4 with the following operation table.

1. Find the order of each element of \(G\).
2. Write down a proper subgroup of \(G\).
3. Is the group \(G\) cyclic? Give a reason for your answer.
4. State suitable values for each of \(a , b , c\) and \(d\) in the case where the operation \(*\) is multiplication of complex numbers.

4 The group \(F = \{ \mathrm { p } , \mathrm { q } , \mathrm { r } , \mathrm { s } , \mathrm { t } , \mathrm { u } \}\) consists of the six functions defined by $$\mathrm { p } ( x ) = x \quad \mathrm { q } ( x ) = 1 - x \quad \mathrm { r } ( x ) = \frac { 1 } { x } \quad \mathrm {~s} ( x ) = \frac { x - 1 } { x } \quad \mathrm { t } ( x ) = \frac { x } { x - 1 } \quad \mathrm { u } ( x ) = \frac { 1 } { 1 - x } ,$$ the binary operation being composition of functions.

1. Show that st \(= \mathrm { r }\) and find ts.
2. Copy and complete the following composition table for \(F\).

   	p	q	r	s	t	u
   p	p	q	r	s	t	u
   q	q	p	s	r	u	t
   r	r	u	p	t	s	q
   s	s	t	q	u	r	p
   t	t	s	u
   u	u	r	t

3. Give the inverse of each element of \(F\).
4. List all the subgroups of \(F\). The group \(M\) consists of \(\left\{ 1 , - 1 , e ^ { \frac { \pi } { 3 } \mathrm { j } } , e ^ { - \frac { \pi } { 3 } \mathrm { j } } , e ^ { \frac { 2 \pi } { 3 } \mathrm { j } } , e ^ { - \frac { 2 \pi } { 3 } \mathrm { j } } \right\}\) with multiplication of complex numbers as its binary operation.
5. Find the order of each element of \(M\). The group \(G\) consists of the positive integers between 1 and 18 inclusive, under multiplication modulo 19.
6. Show that \(G\) is a cyclic group which can be generated by the element 2 .
7. Explain why \(G\) has no subgroup which is isomorphic to \(F\).
8. Find a subgroup of \(G\) which is isomorphic to \(M\).

4 The twelve distinct elements of an abelian multiplicative group \(G\) are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where \(e\) is the identity element, \(a ^ { 6 } = e\) and \(b ^ { 2 } = e\).

1. Show that the element \(a ^ { 2 } b\) has order 6 .
2. Show that \(\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}\) is a subgroup of \(G\).
3. List all the cyclic subgroups of \(G\). You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
4. Determine the order of each of the elements 11, 17 and 19 .
5. Give a cyclic subgroup of \(H\) with order 4.
6. By identifying possible values for the elements \(a\) and \(b\) above, or otherwise, give one example of each of the following:
   (A) a non-cyclic subgroup of \(H\) with order 12,
   (B) a non-cyclic subgroup of \(H\) with order 4.

4 The group \(G = \{ 1,2,3,4,5,6 \}\) has multiplication modulo 7 as its operation. The group \(H = \{ 1,5,7,11,13,17 \}\) has multiplication modulo 18 as its operation.

1. Show that the groups \(G\) and \(H\) are both cyclic.
2. List all the proper subgroups of \(G\).
3. Specify an isomorphism between \(G\) and \(H\). The group \(S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}\) consists of functions with domain \(\{ 1,2,3 \}\) given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1 \\ \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2 \\ \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2 \\ \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1 \\ \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3 \\ \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
4. Show that ad \(= \mathrm { c }\) and find da.
5. Give a reason why \(S\) is not isomorphic to \(G\).
6. Find the order of each element of \(S\).
7. List all the proper subgroups of \(S\).

1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).

1. In each case, write down the smallest possible value of \(n\) :
   1. if \(G\) is cyclic,
   2. if \(G\) has a proper subgroup of order 3,
   3. if \(G\) has at least two elements of order 2 .
   4. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).

8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)\).

1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7 A commutative group \(G\) has order 18. The elements \(a , b\) and \(c\) have orders 2, 3 and 9 respectively.

1. Prove that \(a b\) has order 6 .
2. Show that \(G\) is cyclic.

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.

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.}

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.

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 \(G\).

5 The group \(G\) consists of a set \(S\) together with \(\times _ { 80 }\), the operation of multiplication modulo 80. It is given that \(S\) is the smallest set which contains the element 11 .

1. By constructing the Cayley table for \(G\), determine all the elements of \(S\). The Cayley table for a second group, \(H\), also with the operation \(\times _ { 80 }\), is shown below.

   \cline { 2 - 5 } \multicolumn{1}{c|}{\(\times _ { 80 }\)}	1	9	31	39
   1	1	9	31	39
   9	9	1	39	31
   31	31	39	1	9
   39	39	31	9	1

2. Use the two Cayley tables to explain why \(G\) and \(H\) are not isomorphic.
3. (i) List

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 \(A\)

4

1. In each of the following cases, a set \(G\) and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on \(G\). Determine which, if any, of the other three group axioms are satisfied by ( \(G , \circ\) ) and which, if any, are not satisfied.
   1. \(G = \{ 2 n + 1 : n \in \mathbb { Z } \}\) and \(\circ\) is addition.
   2. \(G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}\) and ∘ is multiplication.
   3. \(G\) is the set of all real numbers and ∘ is multiplication.
2. A group \(M\) consists of eight \(2 \times 2\) matrices under the operation of matrix multiplication. Five of the eight elements of \(M\) are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i } \\ \mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i } \\ - \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$$
   1. Find the other three elements of \(M\). \(( N , * )\) is another group of order 8, with identity element \(e\). You are given that \(N = \langle a , b , c \rangle\) where \(a * a = b * b = c * c = e\).
   2. State whether \(M\) and \(N\) are isomorphic to each other, 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. 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 \(H\).

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 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.

5 The set \(S\) is defined as $$S = \{ A , B , C , D \}$$ where \(A = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right] \quad B = \left[ \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right] \quad C = \left[ \begin{array} { c c } - 1 & 0 \\ 0 & - 1 \end{array} \right] \quad D = \left[ \begin{array} { c c } 0 & 1 \\ - 1 & 0 \end{array} \right]\) The group \(G\) is formed by \(S\) under matrix multiplication.
The group \(H\) is defined as \(H = ( \langle \mathrm { i } \rangle , \times )\), where \(\mathrm { i } ^ { 2 } = - 1\) 5

1. 1. Prove that \(B\) is a generator of \(G\).
      Fully justify your answer.
      5
      1. (ii) Show that \(G \cong H\).
         Fully justify your answer.
         5
   2. 1. Explain why \(H\) has no subgroups of order 3
         Fully justify your answer.
         5
   3. (ii) Find all of the subgroups of \(H\).

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
      1. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\) 9
   2. 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
   3. 1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\) Fully justify your answer.
         [0pt] [2 marks]
         9
   4. (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}

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 \(( S , \bigcirc )\).

4 Let \(S\) be the set \(\{ 16,36,57,76,96 \}\) 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 multiplication table for \(\left( S , \times _ { H } \right)\) and show that it is a group. [You may use the result that \(\times _ { H }\) is associative on \(S\).]
3. Write down all generators of \(\left( S , \times _ { H } \right)\). Let \(\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1\). The surface \(S\) has equation \(z = \mathrm { f } ( x , y )\).
4. (a) Find \(\mathrm { f } _ { x }\) and \(\mathrm { f } _ { y }\).
   (b) Show that f has a stationary point at \(P ( 0,0,1 )\) and find the coordinates of the second stationary point, \(Q\), of \(z\).
5. (a) State the equation of the plane of symmetry of the surface \(S\).
   (b) By considering the intersections of \(S\) at \(P\) with the planes \(y = x\) and \(y = - x\), show that \(P\) is a saddle-point of \(S\). A customer takes out a loan of \(\pounds P\) from a bank at an annual interest rate of \(6.32 \%\). Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of \(\pounds 960\) in order to reduce the outstanding amount of the loan. Let \(L _ { n }\) denote the outstanding amount of the loan at the end of month \(n\) after the fixed payment has been made, with \(L _ { 0 } = P\).
6. Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation $$L _ { n + 1 } = 1.00512 L _ { n } - 960 ,$$ with \(L _ { 0 } = P , n \geq 0\).
7. Solve, in terms of \(n\) and \(P\), the first order recurrence relation given in part (i).
8. Given that the loan is fully repaid after 10 years, evaluate \(P\) (to the nearest whole number).
9. Comment on the practicality of the model.
10. Let \(N = 10 a + b\) and \(M = a - 5 b\) where \(a\) and \(b\) are integers such that \(a \geq 1\) and \(0 \leq b \leq 9\). \(N\) is to be tested for divisibility by 17 .
    (a) Prove that \(17 \mid N\) if and only if \(17 \mid M\).
    (b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that \(17 \mid 4097\).
11. (a) Show that, for \(n \geq 2\), any number of the form \(1001 _ { n }\) is composite.
    (b) Given that \(n\) is a positive even number, provide a counter-example to show that the statement "any number of the form \(10001 _ { n }\) is prime" is false. \section*{Copyright Information:} }{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 Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the \section*{...day June 20XX - Morning/Afternoon} AS Level Further Mathematics A
    Unit Y535 Additional Pure Mathematics SAMPLE MARK SCHEME Duration: 1 hour 15 minutes MAXIMUM MARK 60 \section*{DRAFT} \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for AS Level Further Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
    If you are in any doubt whatsoever you should contact your Team Leader.
    The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
    Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
    d When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
    e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
    Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
    f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question. Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
    h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
    i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader.
    j If in any case the scheme operates with considerable unfairness consult your Team Leader. PS = Problem Solving
    M = Modelling

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.