8.03e Order of elements: and order of groups

5 A multiplicative group \(G\) of order 9 has distinct elements \(p\) and \(q\), both of which have order 3 . The group is commutative, the identity element is \(e\), and it is given that \(q \neq p ^ { 2 }\).

1. Write down the elements of a proper subgroup of \(G\)
   1. which does not contain \(q\),
   2. which does not contain \(p\).
   3. Find the order of each of the elements \(p q\) and \(p q ^ { 2 }\), justifying your answers.
   4. State the possible order(s) of proper subgroups of \(G\).
   5. Find two proper subgroups of \(G\) which are distinct from those in part (i), simplifying the elements.

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.

\(\mathbf { 4 }\) 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 binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.

1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
   (B) Show that \(S\), with the operation \(*\), is a group.
4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.

   \(\circ\)	0	1	2	4	5	6
   0	0	1	2	4	5	6
   1	1	4	0	6	2	5
   2	2	0	5	1	6	4
   4	4	6	1	5	0	2
   5	5	2	6	0	4	1
   6	6	5	4	2	1	0

6. Find the order of each element of \(G\).
7. List all the subgroups of \(G\).

4 The group \(F = \{ \mathrm { p } , \mathrm { q } , \mathrm { r } , \mathrm { s } , \mathrm { t } , \mathrm { u } \}\) consists of the six functions defined by $$\mathrm { p } ( x ) = x \quad \mathrm { q } ( x ) = 1 - x \quad \mathrm { r } ( x ) = \frac { 1 } { x } \quad \mathrm {~s} ( x ) = \frac { x - 1 } { x } \quad \mathrm { t } ( x ) = \frac { x } { x - 1 } \quad \mathrm { u } ( x ) = \frac { 1 } { 1 - x } ,$$ the binary operation being composition of functions.

1. Show that st \(= \mathrm { r }\) and find ts.
2. Copy and complete the following composition table for \(F\).

   	p	q	r	s	t	u
   p	p	q	r	s	t	u
   q	q	p	s	r	u	t
   r	r	u	p	t	s	q
   s	s	t	q	u	r	p
   t	t	s	u
   u	u	r	t

3. Give the inverse of each element of \(F\).
4. List all the subgroups of \(F\). The group \(M\) consists of \(\left\{ 1 , - 1 , e ^ { \frac { \pi } { 3 } \mathrm { j } } , e ^ { - \frac { \pi } { 3 } \mathrm { j } } , e ^ { \frac { 2 \pi } { 3 } \mathrm { j } } , e ^ { - \frac { 2 \pi } { 3 } \mathrm { j } } \right\}\) with multiplication of complex numbers as its binary operation.
5. Find the order of each element of \(M\). The group \(G\) consists of the positive integers between 1 and 18 inclusive, under multiplication modulo 19.
6. Show that \(G\) is a cyclic group which can be generated by the element 2 .
7. Explain why \(G\) has no subgroup which is isomorphic to \(F\).
8. Find a subgroup of \(G\) which is isomorphic to \(M\).

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 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.

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

1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
   1. Prove that the set of real numbers, together with the operation \(*\), forms a group.
   2. State, with a reason, whether the group is commutative.
   3. Prove that there are no elements of order 2.
   4. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.

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.

2 The elements of a group \(G\) are the complex numbers \(a + b \mathrm { i }\) where \(a , b \in \{ 0,1,2,3,4 \}\). These elements are combined under the operation of addition modulo 5 .

1. State the identity element and the order of \(G\).
2. Write down the inverse of \(2 + 4 \mathrm { i }\).
3. Show that every non-zero element of \(G\) has order 5 .

8 A multiplicative group \(H\) has the elements \(\left\{ e , a , a ^ { 2 } , a ^ { 3 } , w , a w , a ^ { 2 } w , a ^ { 3 } w \right\}\) where \(e\) is the identity, elements \(a\) and \(w\) have orders 4 and 2 respectively and \(w a = a ^ { 3 } w\).

1. Show that \(w a ^ { 2 } = a ^ { 2 } w\) and also that \(w a ^ { 3 } = a w\).
2. Hence show that each of \(a w , a ^ { 2 } w\) and \(a ^ { 3 } w\) has order 2 .
3. Find two non-cyclic subgroups of \(H\) of order 4, and show that they are not cyclic.

7 A commutative group \(G\) has order 18. The elements \(a , b\) and \(c\) have orders 2, 3 and 9 respectively.

1. Prove that \(a b\) has order 6 .
2. Show that \(G\) is cyclic.

2 The elements of a group \(G\) are polynomials of the form \(a + b x + c x ^ { 2 }\), where \(a , b , c \in \{ 0,1,2,3,4 \}\). The group operation is addition, where the coefficients are added modulo 5 .

1. State the identity element.
2. State the inverse of \(3 + 2 x + x ^ { 2 }\).
3. State the order of \(G\). The proper subgroup \(H\) contains \(2 + x\) and \(1 + x\).
4. Find the order of \(H\), justifying your answer.

8 A multiplicative group \(Q\) of order 8 has elements \(\left\{ e , p , p ^ { 2 } , p ^ { 3 } , a , a p , a p ^ { 2 } , a p ^ { 3 } \right\}\), where \(e\) is the identity. The elements have the properties \(p ^ { 4 } = e\) and \(a ^ { 2 } = p ^ { 2 } = ( a p ) ^ { 2 }\).

1. Prove that \(a = p a p\) and that \(p = a p a\).
2. Find the order of each of the elements \(p ^ { 2 } , a , a p , a p ^ { 2 }\).
3. Prove that \(\left\{ e , a , p ^ { 2 } , a p ^ { 2 } \right\}\) is a subgroup of \(Q\).
4. Determine whether \(Q\) is a commutative group.

8 A non-commutative multiplicative group \(G\) of order eight has the elements $$\left\{ e , a , a ^ { 2 } , a ^ { 3 } , b , a b , a ^ { 2 } b , a ^ { 3 } b \right\}$$ where \(e\) is the identity and \(a ^ { 4 } = b ^ { 2 } = e\).

1. Show that \(b a \neq a ^ { n }\) for any integer \(n\).
2. Prove, by contradiction, that \(b a \neq a ^ { 2 } b\) and also that \(b a \neq a b\). Deduce that \(b a = a ^ { 3 } b\).
3. Prove that \(b a ^ { 2 } = a ^ { 2 } b\).
4. Construct group tables for the three subgroups of \(G\) of order four. \section*{END OF QUESTION PAPER}

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.

7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.

1. Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
2. Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
3. Write down the order of each element of the group.
4. Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
5. 1. Give an example of a non-cyclic group of order 6 .
   2. Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).

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.

8

1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
2. Prove that any group, in which every (non-identity) element is self-inverse, is abelian.
3. A student believes that, if \(x\) and \(y\) are two distinct, non-identity, self-inverse elements of a group, then the element \(x y\) is also self-inverse. The table shown here is the Cayley table for the non-cyclic group of order 6, having elements \(i , a , b , c , d\) and \(e\), where \(i\) is the identity.

   	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
   \(i\)	\(i\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
   \(a\)	\(a\)	\(i\)	\(d\)	\(e\)	\(b\)	\(c\)
   \(b\)	\(b\)	\(e\)	\(i\)	\(d\)	\(c\)	\(a\)
   \(c\)	\(c\)	\(d\)	\(e\)	\(i\)	\(a\)	\(b\)
   \(d\)	\(d\)	\(c\)	\(a\)	\(b\)	\(e\)	\(i\)
   \(e\)	\(e\)	\(b\)	\(c\)	\(a\)	\(i\)	\(d\)

   By considering the elements of this group, produce a counter-example which proves that this student is wrong.
4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct, non-identity, self-inverse elements of \(G\). By considering the set \(\mathrm { H } = \{ \mathrm { i } , \mathrm { x } , \mathrm { y } , \mathrm { xy } \}\), prove by contradiction that not all elements of \(G\) are self-inverse.

4 A binary operation, ○, is defined on a set of numbers, \(A\), in the following way. \(a \circ b = \mathrm { k } _ { 1 } \mathrm { a } - \mathrm { k } _ { 2 } \mathrm {~b} + \mathrm { k } _ { 3 }\), for \(a , b \in A\),
where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are constants (which are not necessarily in \(A\) ) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning. You are initially given the following information about ○ and \(A\).

• \(A = \mathbb { R }\)
• \(0 \circ 0 = 2\)
• An identity element, \(e\), exists for ∘ in \(A\)

4

1. In each of the following cases, a set \(G\) and a binary operation ∘ are given. The operation ∘ may be assumed to be associative on \(G\). Determine which, if any, of the other three group axioms are satisfied by ( \(G , \circ\) ) and which, if any, are not satisfied.
   1. \(G = \{ 2 n + 1 : n \in \mathbb { Z } \}\) and \(\circ\) is addition.
   2. \(G = \{ a + b \sqrt { 2 } : a , b \in \mathbb { Z } \}\) and ∘ is multiplication.
   3. \(G\) is the set of all real numbers and ∘ is multiplication.
2. A group \(M\) consists of eight \(2 \times 2\) matrices under the operation of matrix multiplication. Five of the eight elements of \(M\) are as follows. $$\frac { 1 } { \sqrt { 2 } } \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } - 1 & \mathrm { i } \\ \mathrm { i } & - 1 \end{array} \right) \quad \frac { 1 } { \sqrt { 2 } } \left( \begin{array} { r r } 1 & - \mathrm { i } \\ - \mathrm { i } & 1 \end{array} \right) \quad \left( \begin{array} { l l } 0 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right) \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$$
   1. Find the other three elements of \(M\). \(( N , * )\) is another group of order 8, with identity element \(e\). You are given that \(N = \langle a , b , c \rangle\) where \(a * a = b * b = c * c = e\).
   2. State whether \(M\) and \(N\) are isomorphic to each other, giving a reason for your answer.