8.03b Cayley tables: construct for finite sets under binary operation

4 Elements of the set \(\{ p , q , r , s , t \}\) are combined according to the operation table shown below.

1. Verify that \(q ( s t ) = ( q s ) t\).
2. Assuming that the associative property holds for all elements, prove that the set \(\{ p , q , r , s , t \}\), with the operation table shown, forms a group \(G\).
3. A multiplicative group \(H\) is isomorphic to the group \(G\). The identity element of \(H\) is \(e\) and another element is \(d\). Write down the elements of \(H\) in terms of \(e\) and \(d\).

\(\mathbf { 4 }\) The group \(G\) consists of the 8 complex matrices \(\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}\) under matrix multiplication, where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0 \\ 0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j } \\ \mathrm { j } & 0 \end{array} \right)$$

1. Copy and complete the following composition table for \(G\).

   	I	J	K	L	-I	-J	-K	\(- \mathbf { L }\)
   I	I	J	K	L	-I	-J	-K	-L
   J	J	-I	L	-K	-J	I	-L	K
   K	K	-L	-I
   L	L	K
   -I	-I	-J
   -J	-J	I
   -K	-K	L
   -L	-L	-K

   (Note that \(\mathbf { J K } = \mathbf { L }\) and \(\mathbf { K J } = - \mathbf { L }\).)
2. State the inverse of each element of \(G\).
3. Find the order of each element of \(G\).
4. Explain why, if \(G\) has a subgroup of order 4, that subgroup must be cyclic.
5. Find all the proper subgroups of \(G\).
6. Show that \(G\) is not isomorphic to the group of symmetries of a square.

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. Show that the set \(P = \{ 1,5,7,11 \}\), under the binary operation of multiplication modulo 12, is a group. You may assume associativity. A group \(Q\) has identity element \(e\). The result of applying the binary operation of \(Q\) to elements \(x\) and \(y\) is written \(x y\), and the inverse of \(x\) is written \(x ^ { - 1 }\).
2. Verify that the inverse of \(x y\) is \(y ^ { - 1 } x ^ { - 1 }\). Three elements \(a , b\) and \(c\) of \(Q\) all have order 2, and \(a b = c\).
3. By considering the inverse of \(c\), or otherwise, show that \(b a = c\).
4. Show that \(b c = a\) and \(a c = b\). Find \(c b\) and \(c a\).
5. Complete the composition table for \(R = \{ e , a , b , c \}\). Hence show that \(R\) is a subgroup of \(Q\) and that \(R\) is isomorphic to \(P\). The group \(T\) of symmetries of a square contains four reflections \(A , B , C , D\), the identity transformation \(E\) and three rotations \(F , G , H\). The binary operation is composition of transformations. The composition table for \(T\) is given below.

   	A	\(B\)	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
   A	E	\(G\)	\(H\)	\(F\)	\(A\)	D	\(B\)	\(C\)
   B	G	E	\(F\)	\(H\)	\(B\)	C	A	D
   C	\(F\)	H	E	G	C	A	D	\(B\)
   D	\(H\)	\(F\)	\(G\)	E	\(D\)	\(B\)	C	\(A\)
   E	A	\(B\)	C	D	\(E\)	\(F\)	\(G\)	\(H\)
   F	C	D	\(B\)	A	\(F\)	G	\(H\)	\(E\)
   \(G\)	B	\(A\)	\(D\)	C	\(G\)	H	E	\(F\)
   \(H\)	D	C	A	B	\(H\)	E	\(F\)	G

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

4

1. The composition table for a group \(G\) of order 8 is given below.

   	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
   \(a\)	\(c\)	\(e\)	\(b\)	\(f\)	\(a\)	\(h\)	\(d\)	\(g\)
   \(b\)	\(e\)	\(c\)	\(a\)	\(g\)	\(b\)	\(d\)	h	\(f\)
   \(c\)	\(b\)	\(a\)	\(e\)	\(h\)	\(c\)	\(g\)	\(f\)	\(d\)
   \(d\)	\(f\)	\(g\)	\(h\)	\(a\)	\(d\)	\(c\)	\(e\)	\(b\)
   \(e\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
   \(f\)	\(h\)	\(d\)	\(g\)	\(c\)	\(f\)	\(b\)	\(a\)	\(e\)
   \(g\)	\(d\)	\(h\)	\(f\)	\(e\)	\(g\)	\(a\)	\(b\)	\(c\)
   \(h\)	\(g\)	\(f\)	\(d\)	\(b\)	\(h\)	\(e\)	\(c\)	\(a\)

   1. State which is the identity element, and give the inverse of each element of \(G\).
   2. Show that \(G\) is cyclic.
   3. Specify an isomorphism between \(G\) and the group \(H\) consisting of \(\{ 0,2,4,6,8,10,12,14 \}\) under addition modulo 16 .
   4. Show that \(G\) is not isomorphic to the group of symmetries of a square.
2. The set \(S\) consists of the functions \(\mathrm { f } _ { n } ( x ) = \frac { x } { 1 + n x }\), for all integers \(n\), and the binary operation is composition of functions.
   1. Show that \(\mathrm { f } _ { m } \mathrm { f } _ { n } = \mathrm { f } _ { m + n }\).
   2. Hence show that the binary operation is associative.
   3. Prove that \(S\) is a group.
   4. Describe one subgroup of \(S\) which contains more than one element, but which is not the whole of \(S\).

7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .

1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right)\).
3. State the order of \(\left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.

4 The elements \(a , b , c , d\) are combined according to the operation table below, to form a group \(G\) of order 4. Group \(G\) is isomorphic either to the multiplicative group \(H = \left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\) or to the multiplicative group \(K = \{ e , p , q , p q \}\). It is given that \(r ^ { 4 } = e\) in group \(H\) and that \(p ^ { 2 } = q ^ { 2 } = e\) in group \(K\), where \(e\) denotes the identity in each group.

1. Write down the operation tables for \(H\) and \(K\).
2. State the identity element of \(G\).
3. Demonstrate the isomorphism between \(G\) and either \(H\) or \(K\) by listing how the elements of \(G\) correspond to the elements of the other group. If the correspondence can be shown in more than one way, list the alternative correspondence(s).

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 \(G\).

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

5

1. The group \(G\) consists of the set \(S = \{ 1,9,17,25 \}\) under \(\times _ { 32 }\), the operation of multiplication modulo 32.
   1. Complete the Cayley table for \(G\) given in the Printed Answer Booklet.
   2. Up to isomorphisms, there are only two groups of order 4.
      • \(C _ { 4 }\), the cyclic group of order 4
      • \(K _ { 4 }\), the non-cyclic (Klein) group of order 4
      State, with justification, to which of these two groups \(G\) is isomorphic.
   3. 1. List the odd quadratic residues modulo 32.
      2. Given that \(n\) is an odd integer, prove that \(n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )\).

1 The following Cayley table is for a set \(\{ a , b , c , d \}\) under a suitable binary operation.

1. Given that the Latin square property holds for this Cayley table, complete it using the table supplied in the Printed Answer Booklet.
2. Using your completed Cayley table, explain why the set does not form a group under the binary operation.

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.

3 Fig. 3.1 shows an equilateral triangle, with vertices \(\mathrm { A } , \mathrm { B }\) and C , and the three axes of symmetry of the triangle, \(\mathrm { S } _ { \mathrm { a } } , \mathrm { S } _ { \mathrm { b } }\) and \(\mathrm { S } _ { \mathrm { c } }\). The axes of symmetry are fixed in space and all intersect at the point O . \begin{figure}[h] \end{figure} There are six distinct transformations under which the image of the triangle is indistinguishable from the triangle itself, ignoring labels.
These are denoted by \(\mathrm { I } , \mathrm { M } _ { a ^ { \prime } } \mathrm { M } _ { \mathrm { b } ^ { \prime } } , \mathrm { M } _ { \mathrm { c } ^ { \prime } } , \mathrm { R } _ { 120 }\) and \(\mathrm { R } _ { 240 }\) where

• I is the identity transformation
• \(\mathrm { M } _ { \mathrm { a } }\) is a reflection in the mirror line \(\mathrm { S } _ { \mathrm { a } }\) (and likewise for \(\mathrm { M } _ { \mathrm { b } }\) and \(\mathrm { M } _ { \mathrm { c } }\) )
• \(\mathrm { R } _ { 120 }\) is an anticlockwise rotation by \(120 ^ { \circ }\) about O (and likewise for \(\mathrm { R } _ { 240 }\) ).

Composition of transformations is denoted by ○.
Fig. 3.2 illustrates the composition of \(R _ { 120 }\) followed by \(R _ { 240 }\), denoted by \(R _ { 240 } \circ R _ { 120 }\). This shows that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 }\) is equivalent to the identity transformation, so that \(\mathrm { R } _ { 240 } \circ \mathrm { R } _ { 120 } = \mathrm { I }\). \begin{figure}[h] \end{figure}

1. Using the blank diagrams in the Printed Answer Booklet, find the single transformation which is equivalent to each of the following.
   The set of the six transformations is denoted by G and you are given that \(( \mathrm { G } , \circ )\) is a group. The table below is a mostly empty composition table for \(\circ\). The entry given is that for \(R _ { 240 } \circ R _ { 120 }\).
   First transformation performed is
   followed by

   ◯	I	\(\mathrm { M } _ { \mathrm { a } }\)	\(\mathrm { M } _ { \mathrm { b } }\)	\(\mathrm { M } _ { \mathrm { c } }\)	\(\mathrm { R } _ { 120 }\)	\(\mathrm { R } _ { 240 }\)
   I
   \(\mathrm { M } _ { \mathrm { a } }\)
   \(\mathrm { M } _ { \mathrm { b } }\)
   \(\mathrm { M } _ { \mathrm { c } }\)
   \(\mathrm { R } _ { 120 }\)
   \(\mathrm { R } _ { 240 }\)					I

2. Complete the copy of this table in the Printed Answer Booklet. You can use some or all of the spare copies of the diagram in the Printed Answer Booklet to help.
3. Explain why there can be no subgroup of \(( \mathrm { G } , \circ )\) of order 4.
4. A student makes the following claim.
   "If all the proper non-trivial subgroups of a group are abelian then the group itself is abelian."
   Explain why the claim is incorrect, justifying your answer fully.
5. With reference to the order of elements in the groups, explain why ( \(\mathrm { G } , \circ\) ) is not isomorphic to \(\mathrm { C } _ { 6 }\), the cyclic group of order 6 .

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.

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]

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 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}