8.03k Lagrange's theorem: order of subgroup divides order of group

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.

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 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}

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.

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.

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

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

1. The group \(\mathrm { S } _ { 4 }\) is the set of all possible permutations that can be performed on the four numbers 1, 2, 3 and 4, under the operation of composition.

For the group \(\mathrm { S } _ { 4 }\)

1. write down the identity element,
2. write down the inverse of the element \(a\), where $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array} \right)$$
3. demonstrate that the operation of composition is associative using the following elements $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array} \right) \quad b = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 2 & 4 & 3 & 1 \end{array} \right) \quad \text { and } c = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \end{array} \right)$$
4. Explain why it is possible for the group \(\mathrm { S } _ { 4 }\) to have a subgroup of order 4 You do not need to find such a subgroup.

4 The group \(G\) consists of the symmetries of the equilateral triangle \(A B C\) under the operation of composition of transformations (which may be assumed to be associative). Three elements of \(G\) are

• \(\boldsymbol { i }\), the identity
• \(\boldsymbol { j }\), the reflection in the vertical line of symmetry of the triangle
• \(\boldsymbol { k }\), the anticlockwise rotation of \(120 ^ { \circ }\) about the centre of the triangle.

These are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762} \includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}

1. Explain why the order of \(G\) is 6 .
2. Determine
   • the order of \(\boldsymbol { j }\),
   • the order of \(\boldsymbol { k }\).
   • - Express, in terms of \(\boldsymbol { j }\) and/or \(\boldsymbol { k }\), each of the remaining three elements of \(G\).
   • Draw a diagram for each of these elements.
   • Is the operation of composition of transformations on \(G\) commutative? Justify your answer.
   • List all the proper subgroups of \(G\).

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 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(\quad x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).

1. Let \(x \in G\). Justify the following statements:
   1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
   2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
   3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
   4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.

6 A group \(G\) has order 12.

1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
   (1) \(x\) has order 6 ,
   (2) \(x ^ { 3 } = y ^ { 2 }\),
   (3) \(x y x = y\).
2. Prove that \(y x ^ { 2 } y = x\).
3. Prove that \(G\) is not a cyclic group.

\(Q\) is a multiplicative group of order 12.

1. Two elements of \(Q\) are \(a\) and \(r\). It is given that \(r\) has order 6 and that \(a^2 = r^3\). Find the orders of the elements \(a\), \(a^2\), \(a^3\) and \(r^2\). [4]

The table below shows the number of elements of \(Q\) with each possible order. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.

1. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). [5]