Isomorphism between groups

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

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

4

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 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 \(S\).

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. 1. List
      • all the proper subgroups of \(G\),
4. all the proper subgroups of \(H\).
   (ii) Use your answers to (c) (i) to give another reason why \(G\) and \(H\) are not isomorphic.

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
2. (ii) Show that \(G \cong H\).
   Fully justify your answer.
   5
3. 1. Explain why \(H\) has no subgroups of order 3
      Fully justify your answer.
      5
4. (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
2. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\)
   9
3. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
4. 1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\)
      Fully justify your answer.
      [0pt] [2 marks]
      9
5. (ii) Determine, with a reason, whether or not \(C \cong V\)
   \(\mathbf { 9 }\) (c) The group \(G\) has order 16
   Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim.
   \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}