8.03l Isomorphism: determine using informal methods

1

1. Show that the set of numbers \(\{ 3,5,7 \}\), under multiplication modulo 8, does not form a group.
2. The set of numbers \(\{ 3,5,7 , a \}\), under multiplication modulo 8 , forms a group. Write down the value of \(a\).
3. State, justifying your answer, whether or not the group in part (ii) is isomorphic to the multiplicative group \(\left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\), where \(e\) is the identity and \(r ^ { 4 } = e\).

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 \(d\).

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 \(S\).

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 \(n\).

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 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. Show that the set \(G = \{ 1,3,4,5,9 \}\), under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
2. Explain why any two groups of order 5 must be isomorphic to each other. The set \(H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}\) is a group under the binary operation of multiplication of complex numbers.
3. Specify an isomorphism between the groups \(G\) and \(H\). The set \(K\) consists of the 25 ordered pairs \(( x , y )\), where \(x\) and \(y\) are elements of \(G\). The set \(K\) is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, \(( 3,5 ) ( 4,4 ) = ( 1,9 )\).
4. Write down the identity element of \(K\), and find the inverse of the element \(( 9,3 )\).
5. Explain why \(( x , y ) ^ { 5 } = ( 1,1 )\) for every element \(( x , y )\) in \(K\).
6. Deduce that all the elements of \(K\), except for one, have order 5. State which is the exceptional element.
7. A subgroup of \(K\) has order 5 and contains the element (9, 3). List the elements of this subgroup.
8. Determine how many subgroups of \(K\) there are with order 5 .

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.}

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 \(n ^ { 6 } + 3 n ^ { 4 } + 7 n ^ { 2 } \equiv 11 ( \bmod 32 )\).

8 The group \(G\) is cyclic and of order 12.

1. 1. State the possible orders of all the proper subgroups of \(G\). You must justify your answers.
   2. List all the elements of each of these subgroups.
   3. Explain why \(G\) must be abelian. The group \(\mathbb { Z } _ { k }\) is the cyclic group of order \(k\), consisting of the elements \(\{ 0,1,2 , \ldots , k - 1 \}\) under the operation \(+ _ { k }\) of addition modulo \(k\). The coordinate group \(\mathrm { C } _ { \mathrm { mn } }\) is the group which consists of elements of the form \(( x , y )\), where \(\mathrm { x } \in \mathbb { Z } _ { \mathrm { m } }\) and \(\mathrm { y } \in \mathbb { Z } _ { \mathrm { n } }\), under the operation \(\oplus\) given by \(\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) \oplus \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right) = \left( \mathrm { x } _ { 1 } + { } _ { \mathrm { m } } \mathrm { x } _ { 2 } , \mathrm { y } _ { 1 } + { } _ { \mathrm { n } } \mathrm { y } _ { 2 } \right)\). For example, for \(m = 5\) and \(n = 2 , ( 3,0 ) \oplus ( 4,1 ) = ( 2,1 )\).
2. 1. List all the elements of \(\mathrm { J } = \mathrm { C } _ { 34 }\).
   2. Show that \(G\) and \(J\) are isomorphic. There is a second coordinate group of order 12; that is, \(\mathrm { K } = \mathrm { C } _ { \mathrm { mn } }\), where \(1 < \mathrm { m } < \mathrm { n } < 12\) but neither \(m\) nor \(n\) is equal to 3 or 4 .
3. 1. State the values of \(m\) and \(n\) which give \(K\).
   2. Hence list all of the elements of \(K\).
   3. Explain why \(K\) must be abelian.
4. Show that \(G\) and \(K\) are not isomorphic. \section*{END OF QUESTION PAPER}

6 The group \(G\) consists of the set \(\{ 3,6,9,12,15,18,21,24,27,30,33,36 \}\) under \(\times _ { 39 }\), the operation of multiplication modulo 39.

1. List the possible orders of proper subgroups of \(G\), justifying your answer.
2. List the elements of the subset of \(G\) generated by the element 3 .
3. State the identity element of \(G\).
4. Determine the order of the element 18 .
5. Find the two elements \(g _ { 1 }\) and \(g _ { 2 }\) in \(G\) which satisfy \(g \times { } _ { 39 } g = 3\). The group \(H\) consists of the set \(\{ 1,2,3,4,5,6,7,8,9,10,11,12 \}\) under \(\times _ { 13 }\), the operation of multiplication modulo 13. You are given that \(G\) is isomorphic to \(H\). A student states that \(G\) is isomorphic to \(H\) because each element \(3 x\) in \(G\) maps directly to the element \(x\) in \(H\) (for \(x = 1,2,3,4,5,6,7,8,9,10,11,12\) ).
6. Explain why this student is incorrect.

2 The following Cayley table is for \(G\), a group of order 6. The identity element is \(e\) and the group is generated by the elements \(a\) and \(b\).

1. List all the proper subgroups of \(G\).
2. State another group of order 6 to which \(G\) is isomorphic.

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 \(H\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).

1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
3. Explain why each of the following statements is false, making your reasoning clear.
   1. \(G\) has a subgroup of order 4
   2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
   \section*{Diagram 1} \section*{Spare copy of Diagram 1}
   \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
   Only use this diagram if you need to redraw your answer to part (d).

1. The set of matrices \(G = \{ \mathbf { I } , \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E } \}\) where

$$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) \quad \mathbf { D } = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right) \quad \mathbf { E } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 1 \end{array} \right)$$ with the operation \(\otimes _ { 2 }\) of matrix multiplication with entries evaluated modulo 2 , forms a group.

1. Show that \(\mathbf { B }\) is an element of order 3 in \(G\).
2. Determine the orders of the other elements of \(G\).
3. Give a reason why \(G\) is not isomorphic to
   1. a cyclic group of order 6
   2. the group of symmetries of a regular hexagon. The group \(H\) of permutations of the numbers 1, 2 and 3 contains the following elements, denoted in two-line notation, $$\begin{array} { l l l } e = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 2 & 3 \end{array} \right) & a = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} \right) & b = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 3 & 1 & 2 \end{array} \right) \\ c = \left( \begin{array} { l l } 1 & 2 \\ 1 & 3 \\ 2 \end{array} \right) & d = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \end{array} \right) & f = \left( \begin{array} { l l } 1 & 2 \\ 3 & 2 \end{array} \right) \end{array}$$
4. Determine an isomorphism between the groups \(G\) and \(H\).

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 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 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}