Subgroups and cosets

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.

9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)

1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
   1. The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
   2. The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
   3. The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 4

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.

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.

8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)\).

1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

8 Let \(G\) be any multiplicative group. \(H\) is a subset of \(G\). \(H\) consists of all elements \(h\) such that \(h g = g h\) for every element \(g\) in \(G\).

1. Prove that \(H\) is a subgroup of \(G\). Now consider the case where \(G\) is given by the following table:

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

2. Show that \(H\) consists of just the identity element.

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. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
2. Show that \(M\) is cyclic.
3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.

   	A	B	C	D	E	F	G	H	I	J
   A	C	J	G	H	A	B	I	F	E	D
   B	F	E	H	G	B	A	D	C	J	I
   C	G	D	I	F	C	J	E	B	A	H
   D	J	C	B	E	D	G	F	I	H	A
   E	A	B	C	D	E	F	G	H	I	J
   F	H	I	D	C	F	E	J	A	B	G
   G	I	H	E	B	G	D	A	J	C	F
   H	D	G	J	A	H	I	B	E	F	C
   I	E	F	A	J	I	H	C	D	G	B
   J	B	A	F	I	J	C	H	G	D	E

   One of these transformations is the identity transformation, some are rotations and the rest are reflections.
4. Identify which transformation is the identity, which are rotations and which are reflections.
5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
6. Find the order of each element of \(P\).
7. List all the proper subgroups of \(P\).

8 The group \(G\) is cyclic and of order 12.

1. 1. State the possible orders of all the proper subgroups of \(G\). You must justify your answers.
   2. List all the elements of each of these subgroups.
   3. Explain why \(G\) must be abelian. The group \(\mathbb { Z } _ { k }\) is the cyclic group of order \(k\), consisting of the elements \(\{ 0,1,2 , \ldots , k - 1 \}\) under the operation \(+ _ { k }\) of addition modulo \(k\). The coordinate group \(\mathrm { C } _ { \mathrm { mn } }\) is the group which consists of elements of the form \(( x , y )\), where \(\mathrm { x } \in \mathbb { Z } _ { \mathrm { m } }\) and \(\mathrm { y } \in \mathbb { Z } _ { \mathrm { n } }\), under the operation \(\oplus\) given by \(\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) \oplus \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right) = \left( \mathrm { x } _ { 1 } + { } _ { \mathrm { m } } \mathrm { x } _ { 2 } , \mathrm { y } _ { 1 } + { } _ { \mathrm { n } } \mathrm { y } _ { 2 } \right)\). For example, for \(m = 5\) and \(n = 2 , ( 3,0 ) \oplus ( 4,1 ) = ( 2,1 )\).
2. 1. List all the elements of \(\mathrm { J } = \mathrm { C } _ { 34 }\).
   2. Show that \(G\) and \(J\) are isomorphic. There is a second coordinate group of order 12; that is, \(\mathrm { K } = \mathrm { C } _ { \mathrm { mn } }\), where \(1 < \mathrm { m } < \mathrm { n } < 12\) but neither \(m\) nor \(n\) is equal to 3 or 4 .
3. 1. State the values of \(m\) and \(n\) which give \(K\).
   2. Hence list all of the elements of \(K\).
   3. Explain why \(K\) must be abelian.
4. Show that \(G\) and \(K\) are not isomorphic. \section*{END OF QUESTION PAPER}

2 The following Cayley table is for \(G\), a group of order 6. The identity element is \(e\) and the group is generated by the elements \(a\) and \(b\).

1. List all the proper subgroups of \(G\).
2. State another group of order 6 to which \(G\) is isomorphic.

1. (i) Let \(G\) be a group of order 5291848

Without performing any division, use proof by contradiction to show that \(G\) cannot have a subgroup of order 11
(ii) (a) Complete the following Cayley table for the set \(X = \{ 2,4,8,14,16,22,26,28 \}\) with the operation of multiplication modulo 30 A copy of this table is given on page 11 if you need to rewrite your Cayley table.
(b) Hence determine whether the set \(X\) with the operation of multiplication modulo 30 forms a group.
[0pt] [You may assume multiplication modulo \(n\) is an associative operation.] Only use this grid if you need to rewrite your Cayley table. (Total for Question 3 is 9 marks)

1. Let \(G\) be a group of order \(46 ^ { 46 } + 47 ^ { 47 }\)

Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of \(G\)

1. 11
2. 21

10 A group \(G\) has distinct elements \(e , a , b , c , \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation.

1. Prove that if \(a \circ a = b\) and \(b \circ b = a\), then the set of elements \(\{ e , a , b \}\) forms a subgroup of \(G\).
2. Prove that if \(a \circ a = b , b \circ b = c\) and \(c \circ c = a\), then the set of elements \(\{ e , a , b , c \}\) does not form a subgroup of \(G\).

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, [1]
   2. if \(G\) has a proper subgroup of order 3, [1]
   3. if \(G\) has at least two elements of order 2. [1]
2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\). [2]

A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if $$a \circ a = b, \quad b \circ b = a$$ then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5] Prove that if $$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$ then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]