8.03h Generators: of cyclic and non-cyclic groups

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

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 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 \(S\).

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.}

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.

3 A non-commutative group \(G\) consists of the six elements \(\left\{ e , a , a ^ { 2 } , b , a b , b a \right\}\) where \(e\) is the identity element, \(a\) is an element of order 3 and \(b\) is an element of order 2 .
By considering the row in \(G\) 's group table in which each of the above elements is pre-multiplied by \(b\), show that \(b a ^ { 2 } = a b\).

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 \(\left( S , \times _ { H } \right)\).

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

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 set \(\{ 1,2,4,8,9,13,15,16 \}\) forms a group under the operation of multiplication modulo 17. Which of the following is a generator of the group? Circle your answer.
[0pt] [1 mark] 491316

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.

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 \(( S , \bigcirc )\).

The group \(G\) has binary operation \(*\) and order \(p\), where \(p\) is a prime number.

1. Determine the number of distinct subgroups of \(G\) Fully justify your answer. [2 marks]
2. \(G\) contains an element \(g\) which has period \(p\)
   1. State the general name given to elements such as \(g\) [1 mark]
   2. State the name of a group that is isomorphic to \(G\) [1 mark]
3. \(G\) contains an element \(g^r\), where \(r < p\) Find, in terms of \(g\), \(r\) and \(p\), the inverse of \(g^r\) [2 marks]
4. In the case when \(p = 5\) and the binary operation \(*\) represents addition modulo 5, \(G\) contains the elements 0, 1, 2, 3 and 4
   1. Explain why \(G\) is closed. [1 mark]
   2. Complete the Cayley table for \((G, *)\) [1 mark]

      \(*\)

\(G\) is a group of order 8.

1. Explain why there is no subgroup of \(G\) of order 6. [1]

You are now given that \(G\) is a cyclic group with the following features: • \(e\) is the identity element of \(G\), • \(g\) is a generator of \(G\), • \(H\) is the subgroup of \(G\) of order 4.

1. Write down the possible generators of \(H\). [2]

\(M\) is the group \((\{0, 1, 2, 3, 4, 5, 6, 7\}, +_8)\) where \(+_8\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).

1. Specify all possible isomorphisms between \(M\) and \(G\). [4]

The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.

1. Show that \(G\) is a cyclic group generated by the element 4. [3]
2. Find another generator for \(G\). Justify your answer. [2]
3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]

A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.

1. Explain why the suggestion by Student P might not work. [2]

Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.

1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]

Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).

1. 1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
   2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
   3. Find a different function \(f\) that will satisfy the requirements of the task. [4]