8.03g Cyclic groups: meaning of the term

38 questions

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Pre-U Pre-U 9795/1 2016 June Q8
12 marks Challenging +1.2
8
  1. \(S\) is the set \(\{ 1,2,4,8,16,32 \}\) and \(\times _ { 63 }\) is the operation of multiplication modulo 63 .
    1. Construct the multiplication table for \(\left( S , \times _ { 63 } \right)\).
    2. Show that \(\left( S , \times _ { 63 } \right)\) forms a group, \(G\). (You may assume that \(\times _ { 63 }\) is associative.)
    3. The group \(H\), also of order 6, has identity element \(e\) and contains two further elements \(x\) and \(y\) with the properties $$x ^ { 2 } = y ^ { 3 } = e \quad \text { and } \quad x y x = y ^ { 2 } .$$ (a) Construct the group table of \(H\).
      (b) List all the proper subgroups of \(H\).
    4. State, with justification, whether \(G\) and \(H\) are isomorphic.
Pre-U Pre-U 9795/1 2016 Specimen Q6
7 marks Challenging +1.8
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.
Pre-U Pre-U 9795/1 2016 Specimen Q8
Challenging +1.2
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where \(p\) is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.
Pre-U Pre-U 9795/1 2017 June Q11
13 marks Standard +0.3
11
  1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
    (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
  2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
  3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
    (b) Determine the value of \(n\).
    (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).
Pre-U Pre-U 9795/1 2019 Specimen Q8
5 marks Challenging +1.8
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .
Pre-U Pre-U 9795/1 2020 Specimen Q8
5 marks Challenging +1.8
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
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 .
Pre-U Pre-U 9795/1 Specimen Q7
9 marks Challenging +1.8
7 The multiplicative group \(G\) has eight elements \(e , a , b , c , a b , a c , b c , a b c\), where \(e\) is the identity. The group is commutative, and the order of each of the elements \(a , b , c\) is 2 .
  1. Find four subgroups of \(G\) of order 4.
  2. Give a reason why no group of order 8 can have a subgroup of order 3 . The group \(H\) has elements \(0,1,2 , \ldots , 7\) with group operation addition modulo 8 .
  3. Find the order of each element of \(H\).
  4. Determine whether \(G\) and \(H\) are isomorphic and justify your conclusion.
OCR FP3 Q1
5 marks Challenging +1.2
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]
OCR FP3 2010 January Q8
12 marks Challenging +1.2
The function f is defined by \(\text{f} : x \mapsto \frac{1}{2-2x}\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\). The function g is defined by \(\text{g}(x) = \text{ff}(x)\).
  1. Show that \(\text{g}(x) = \frac{1-x}{1-2x}\) and that \(\text{gg}(x) = x\). [4]
It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where \(\text{e} : x \mapsto x\) for \(x \in \mathbb{R}, x \neq 0, x \neq \frac{1}{2}, x \neq 1\).
  1. State the orders of the elements f and g. [2]
  2. The inverse of the element f is denoted by h. Find \(\text{h}(x)\). [2]
  3. Construct the operation table for the elements e, f, g, h of the group \(K\). [4]
OCR FP3 2011 June Q6
10 marks Challenging +1.2
  1. The set of polynomials \(\{ax + b\}\), where \(a, b \in \mathbb{R}\), is denoted by \(P\). Assuming that the associativity property holds, prove that \(P\), under addition, is a group. [4]
  2. The set of polynomials \(\{ax + b\}\), where \(a, b \in \{0, 1, 2\}\), is denoted by \(Q\). It is given that \(Q\), under addition modulo 3, is a group, denoted by \((Q, +(\text{mod}3))\).
    1. State the order of the group. [1]
    2. Write down the inverse of the element \(2x + 1\). [1]
    3. \(q(x) = ax + b\) is any element of \(Q\) other than the identity. Find the order of \(q(x)\) and hence determine whether \((Q, +(\text{mod}3))\) is a cyclic group. [4]
AQA Further Paper 3 Discrete 2022 June Q7
8 marks Standard +0.3
The group \(G\) has binary operation \(*\) and order \(p\), where \(p\) is a prime number.
  1. Determine the number of distinct subgroups of \(G\) Fully justify your answer. [2 marks]
  2. \(G\) contains an element \(g\) which has period \(p\)
    1. State the general name given to elements such as \(g\) [1 mark]
    2. State the name of a group that is isomorphic to \(G\) [1 mark]
  3. \(G\) contains an element \(g^r\), where \(r < p\) Find, in terms of \(g\), \(r\) and \(p\), the inverse of \(g^r\) [2 marks]
  4. In the case when \(p = 5\) and the binary operation \(*\) represents addition modulo 5, \(G\) contains the elements 0, 1, 2, 3 and 4
    1. Explain why \(G\) is closed. [1 mark]
    2. Complete the Cayley table for \((G, *)\) [1 mark]
      \(*\)
Pre-U Pre-U 9795/1 2018 June Q10
10 marks Challenging +1.8
  1. Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
  2. Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
  3. The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by $$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$ where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
    1. List the elements of \(G_2\) and state the order of each element. [3]
    2. State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]