Order of elements and cyclic structure

Questions asking to find or determine the order of specific elements, whether the group is cyclic, or which elements generate the group.

5 questions · Standard +0.8

8.03a Binary operations: and their properties on given sets8.03g Cyclic groups: meaning of the term
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OCR FP3 Specimen Q2
6 marks Standard +0.8
2 The set \(S = \{ a , b , c , d \}\) under the binary operation * forms a group \(G\) of order 4 with the following operation table.
\(*\)\(a\)\(b\)\(c\)\(d\)
\(a\)\(d\)\(a\)\(b\)\(c\)
\(b\)\(a\)\(b\)\(c\)\(d\)
\(c\)\(b\)\(c\)\(d\)\(a\)
\(d\)\(c\)\(d\)\(a\)\(b\)
  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.
OCR FP3 2013 June Q7
7 marks Challenging +1.8
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.
AQA Further Paper 3 Discrete Specimen Q2
1 marks Moderate -0.5
2 The set \(\{ 1,2,4,8,9,13,15,16 \}\) forms a group under the operation of multiplication modulo 17. Which of the following is a generator of the group? Circle your answer.
[0pt] [1 mark] 491316
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]
      \(*\)
OCR MEI Further Extra Pure Specimen Q1
10 marks Challenging +1.8
The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.
  1. Show that \(G\) is a cyclic group generated by the element 4. [3]
  2. Find another generator for \(G\). Justify your answer. [2]
  3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]