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

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.

4

1. The binary operation is defined on \(\mathbb { Z }\) by \(a\) b \(b = a + b - a b\) for all \(a , b \in \mathbb { Z }\). Prove that is associative on \(\mathbb { Z }\). The operation ∘ is defined on the set \(A = \{ 0,2,3,4,5,6 \}\) by \(a \circ b = a + b - a b ( \bmod 7 )\) for all \(a , b \in A\).
2. Complete the Cayley table for \(\left( A , { } ^ { \circ } \right)\) given in the Printed Answer Booklet.
3. Prove that \(( A , \circ )\) is a group. You may assume that the operation is associative.
4. List all the subgroups of \(( A , \circ )\).

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

1. State the definition of a bipartite graph. 4
2. A jazz quintet has five musical instruments: bassoon, clarinet, flute, oboe and violin. Jay, Kay, Lee, Mel and Nish are musicians and each plays a musical instrument in the jazz quintet. Jay knows how to play the bassoon and the clarinet.
   Kay knows how to play the bassoon, the oboe and the violin.
   Lee knows how to play the clarinet and the flute.
   Mel knows how to play the clarinet, the oboe and the violin.
   Nish knows how to play the flute, the oboe and the violin. 4 (b) (i) Draw a graph to show which musicians know how to play which instruments. 4 (b) (ii) Nish arrives late to a jazz quintet rehearsal. Each of the other four musicians is already playing an instrument: \begin{displayquote} Jay is playing the clarinet
   Kay is playing the oboe
   Lee is playing the flute
   Mel is playing the violin. \end{displayquote} Explain how the graph in part (b)(i) shows that there is no instrument available that Nish knows how to play. 4 (b) (iii) When Nish arrives the rehearsal stops. When they restart the rehearsal, Nish is playing the flute. Draw all possible subgraphs of the graph in part (b)(i) that show how Jay, Kay, Lee and Mel can each be assigned a unique musical instrument they know how to play.
   [0pt] [2 marks]

5

1. Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
   \end{figure} 5
2. The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for \(S\) under the commutative binary operation • \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2}

   •	\(a\)	\(b\)	\(c\)	\(d\)
   \(a\)	\(b\)	\(a\)	\(a\)	\(c\)
   \(b\)		\(c\)		\(c\)
   \(c\)		\(d\)	\(d\)
   \(d\)			\(d\)	\(d\)

   \end{table} 5 (b) (i) Complete the Cayley table in Figure 2. 5 (b) (ii) Determine whether the binary operation • is associative when acting on the elements of \(S\). Fully justify your answer.

8 The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows a Cayley table for \(S\) under the commutative binary operation \begin{table}[h] \end{table} 8

1. 1. Prove that there exists an identity element for \(S\) under the binary operation
      [0pt] [2 marks]
      8
      1. (ii) State the inverse of \(b\) under the binary operation
         8
   2. Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3}

      \(\times _ { 4 }\)	0	1	2	3
      0	0	0	0	0
      1	0	1	2	3
      2	0	2	0	2
      3	0	3	2	1

      \end{table} Mali says that, by substituting suitable distinct values for \(a , b , c\) and \(d\), the Cayley table in Figure 2 could represent multiplication modulo 4 Use your answers to part (a) to show that Mali's statement is incorrect. \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-20_2491_1736_219_139}

5

1. The set \(S\) is defined as \(S = \{ 0,1,2,3,4,5 \}\) 5
   1. (i) State the identity element of \(S\) under the operation multiplication modulo 6 5
   2. (ii) An element \(g\) of a set is said to be self-inverse under a binary operation * if $$g * g = e$$ where \(e\) is the identity element of the set. Find all the self-inverse elements in \(S\) under the operation multiplication modulo 6
      5
   3. \(\quad\) The set \(T\) is defined as $$T = \{ a , b , c \}$$ Figure 1 shows a partially completed Cayley table for \(T\) under the commutative binary operation - \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1}

      -	\(a\)	\(b\)	c
      \(a\)	\(a\)	c	b
      \(b\)		\(b\)	\(a\)
      c			c

      \end{table} 5
   4. 1. Complete the Cayley table in Figure 1 5
   5. (ii) Prove that is not associative when acting on the elements of \(T\)

3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b \\ & = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e \\ & q \boldsymbol { \Delta } q = e \\ & r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6

1. 1. State the order of \(G\). 6
      1. (ii) Prove that the inverse of \(q r\) is \(q r\).
         6
   2. Complete the Cayley table for elements of \(G\). 6
   3. Complete the Cayley table for elements of \(G\).

      A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
      \(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
      \(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
      \(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
      \(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
      \(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
      \(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

      6
   4. State the name of a group which is isomorphic to \(G\).

5

1. Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
   5
2. In the multiplicative group of integers modulo 13, the group \(G\) is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5 (b) (i) Explain why \(G\) is an abelian group.
   5 (b) (ii) Find the order of \(G\).
   5
3. State the identity element of \(G\) and prove it is an identity element. Fully justify your answer.
   5
4. Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13

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}

6 The set \(S\) consists of all real numbers except 1. The binary operation * is defined for all \(a , b\) in \(S\) by $$a * b = a + b - a b$$

1. By considering the identity \(a + b - a b \equiv 1 - ( a - 1 ) ( b - 1 )\), or otherwise, show that \(S\) is closed under *.
2. Show that * is associative on \(S\).
3. Find the identity of \(S\) under \(*\), and the inverse of \(x\) for all \(x \in S\).
4. The set \(S\), together with the binary operation *, forms a group \(G\). Find a subgroup of \(G\) of order 2 .

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

	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(t\)	\(s\)	\(p\)	\(r\)	\(q\)
\(q\)	\(s\)	\(p\)	\(q\)	\(t\)	\(r\)
\(r\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(s\)	\(r\)	\(t\)	\(s\)	\(q\)	\(p\)
\(t\)	\(q\)	\(r\)	\(t\)	\(p\)	\(s\)

1. Verify that \(q(st) = (qs)t\). [2]
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 \(G\). [4]
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 \(d\). [2]

The operation \(*\) is defined on the elements \((x, y)\), where \(x, y \in \mathbb{R}\), by $$(a, b) * (c, d) = (ac, ad + b).$$ It is given that the identity element is \((1, 0)\).

1. Prove that \(*\) is associative. [3]
2. Find all the elements which commute with \((1, 1)\). [3]
3. It is given that the particular element \((m, n)\) has an inverse denoted by \((p, q)\), where $$(m, n) * (p, q) = (p, q) * (m, n) = (1, 0).$$ Find \((p, q)\) in terms of \(m\) and \(n\). [2]
4. Find all self-inverse elements. [3]
5. Give a reason why the elements \((x, y)\), under the operation \(*\), do not form a group. [1]

A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}

1. Give a reason why \(D\) is not commutative. [1]
2. Write down the orders of any possible proper subgroups of \(D\). [2]
3. List the elements of a proper subgroup which contains
   1. the element \(a\), [1]
   2. the element \(r\). [1]
4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]

The binary operation \(\oplus\) acts on the positive integers \(x\) and \(y\) such that $$x \oplus y = x + y + 8 \pmod{k^2 - 16k + 74}$$ where \(k\) is a positive integer.

1. 1. Show that \(\oplus\) is commutative. [1 mark]
   2. Determine whether or not \(\oplus\) is associative. Fully justify your answer. [2 marks]
2. Find the values of \(k\) for which 3 is an identity element for the set of positive integers under \(\oplus\) [3 marks]

\(T\) is the set \(\{1, 2, 3, 4\}\). A binary operation \(\bullet\) is defined on \(T\) such that \(a \bullet a = 2\) for all \(a \in T\). It is given that \((T, \bullet)\) is a group.

1. Deduce the identity element in \(T\), giving a reason for your answer. [2]
2. Find the value of \(1 \bullet 3\), showing how the result is obtained. [3]
3. 1. Complete a group table for \((T, \bullet)\). [2]
   2. State with a reason whether the group is abelian. [1]

The group \(G\), of order 8, consists of the elements \(\{e, a, b, c, ab, bc, ca, abc\}\), together with a multiplicative binary operation, where \(e\) is the identity and $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$

1. Construct the group table of \(G\). [You are not required to show how individual elements of the table are determined.] [4]
2. List all the proper subgroups of \(G\). [5]