Complete or analyse Cayley table

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 group \(G\), of order 8, is generated by the elements \(a , b , c . G\) has the properties $$a ^ { 2 } = b ^ { 2 } = c ^ { 2 } = e , \quad a b = b a , \quad b c = c b , \quad c a = a c ,$$ where \(e\) is the identity.

1. Using these properties and basic group properties as necessary, prove that \(a b c = c b a\). The operation table for \(G\) is shown below.

   	\(e\)	\(a\)	\(b\)	\(c\)	\(b c\)	ca	\(a b\)	\(a b c\)
   \(e\)	\(e\)	\(a\)	\(b\)	\(c\)	\(b c\)	ca	\(a b\)	\(a b c\)
   \(a\)	\(a\)	\(e\)	\(a b\)	ca	\(a b c\)	\(c\)	\(b\)	\(b c\)
   \(b\)	\(b\)	\(a b\)	\(e\)	\(b c\)	\(c\)	\(a b c\)	\(a\)	ca
   c	\(c\)	ca	\(b c\)	\(e\)	\(b\)	\(a\)	\(a b c\)	\(a b\)
   \(b c\)	\(b c\)	\(a b c\)	\(c\)	\(b\)	\(e\)	\(a b\)	ca	\(a\)
   ca	ca	c	\(a b c\)	\(a\)	\(a b\)	\(e\)	\(b c\)	\(b\)
   \(a b\)	\(a b\)	\(b\)	\(a\)	\(a b c\)	ca	bc	\(e\)	\(c\)
   \(a b c\)	\(a b c\)	\(b c\)	ca	\(a b\)	\(a\)	\(b\)	\(c\)	\(e\)

2. List all the subgroups of order 2 .
3. List five subgroups of order 4.
4. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic.

4 The set \(S\) is defined as \(S = \{ 1,2,3,4 \}\) 4

1. Complete the Cayley Table shown below for \(S\) under the binary operation multiplication modulo 5

   \(\times _ { 5 }\)	1	2	3	4
   1
   2
   3
   4

   4
2. State the identity element for \(S\) under multiplication modulo 5 4
3. State the self-inverse elements of \(S\) under multiplication modulo 5

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.

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
2. \(K _ { 4 }\), 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.

3 The matrix \(\mathbf { A }\) is \(\left( \begin{array} { r r r } - 1 & 2 & 4 \\ 0 & - 1 & - 25 \\ - 3 & 5 & - 1 \end{array} \right)\). Use the Cayley-Hamilton theorem to find \(\mathbf { A } ^ { - 1 }\).
\(4 T\) is the set \(\{ 1,2,3,4 \}\). A binary operation • is defined on \(T\) such that \(a \cdot a = 2\) for all \(a \in T\). It is given that ( \(T , \cdot\) ) is a group.

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

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.

   *	\(e\)	\(\boldsymbol { p }\)	\(\boldsymbol { q }\)	\(r\)	\(s\)
   \(e\)
   \(\boldsymbol { p }\)
   \(\boldsymbol { q }\)
   \(\boldsymbol { r }\)
   \(S\)

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
3. 1. Complete the Cayley table in Figure 2. 5
4. (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
2. (ii) State the inverse of \(b\) under the binary operation
   8
3. 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}

2 The set \(S\) is given by \(S = \{ 0,2,4,6 \}\) 2

1. Construct a Cayley table, using the grid below, for \(S\) under the binary operation addition modulo 8
   \includegraphics[max width=\textwidth, alt={}, center]{18ce34aa-e4c3-4a84-a36d-6542d2319bf5-03_561_563_607_831} 2
2. State the identity element for \(S\) under the binary operation addition modulo 8

6 The set \(S\) is given by \(S = \{ \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } \}\) where
\(\mathbf { A } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right]\)
\(\mathbf { B } = \left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\)
\(\mathbf { C } = \left[ \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right]\)
\(\mathbf { D } = \left[ \begin{array} { l l } 0 & 0 \\ 0 & 0 \end{array} \right]\) 6

1. Complete the Cayley table for \(S\) under matrix multiplication.

   	A	B	C	D
   A	A		D
   B		B
   C			C
   D				D

   6
2. Using the Cayley table above, explain why \(\mathbf { B }\) is the identity element of \(S\) under matrix multiplication.
   [0pt] [1 mark] 6
3. Sam states that the Cayley table in part (a) shows that matrix multiplication is commutative. Comment on the validity of Sam's statement.

5

1. The set \(S\) is defined as \(S = \{ 0,1,2,3,4,5 \}\) 5
2. 1. State the identity element of \(S\) under the operation multiplication modulo 6 5
3. (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
4. \(\quad\) 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
   \(b\)		\(b\)	\(а\)
   c			c

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

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
2. (ii) Prove that the inverse of \(q r\) is \(q r\).
   6
3. Complete the Cayley table for elements of \(G\). 6
4. 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
5. State the name of a group which is isomorphic to \(G\).