Verify group axioms

A question is this type if and only if it asks to prove or verify that a given set with an operation forms a group by checking closure, associativity, identity, and inverses.

32 questions · Standard +0.8

8.03a Binary operations: and their properties on given sets
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OCR FP3 Q2
5 marks Standard +0.3
It is given that the set of complex numbers of the form \(re^{i\theta}\) for \(-\pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.
  1. Write down the inverse of \(5e^{3\pi i}\). [1]
  2. Prove the closure property for the group. [2]
  3. \(Z\) denotes the element \(e^{i\gamma}\), where \(\frac{1}{2}\pi < \gamma < \pi\). Express \(Z^2\) in the form \(e^{i\theta}\), where \(-\pi < \theta \leqslant 0\). [2]
OCR FP3 2010 January Q2
6 marks Challenging +1.2
\(H\) denotes the set of numbers of the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.
  1. Show that the product of any two members of \(H\) is a member of \(H\). [2] It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
  2. State the identity element of the group. [1]
  3. Find the inverse of \(a + b\sqrt{5}\). [2]
  4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse. [1]
OCR FP3 2006 June Q1
5 marks Moderate -0.8
  1. For the infinite group of non-zero complex numbers under multiplication, state the identity element and the inverse of \(1 + 2\mathrm{i}\), giving your answers in the form \(a + ib\). [3]
  2. For the group of matrices of the form \(\begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}\) under matrix addition, where \(a \in \mathbb{R}\), state the identity element and the inverse of \(\begin{pmatrix} 3 & 0 \\ 0 & 0 \end{pmatrix}\). [2]
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 2024 June Q1
1 marks Moderate -0.5
Which one of the following sets forms a group under the given binary operation? Tick \((\checkmark)\) one box. [1 mark]
SetBinary Operation
\(\{1, 2, 3\}\)Addition modulo 4\(\square\)
\(\{1, 2, 3\}\)Multiplication modulo 4\(\square\)
\(\{0, 1, 2, 3\}\)Addition modulo 4\(\square\)
\(\{0, 1, 2, 3\}\)Multiplication modulo 4\(\square\)
OCR MEI Further Extra Pure 2019 June Q6
13 marks Challenging +1.8
  1. Given that \(\sqrt{7}\) is an irrational number, prove that \(a^2 - 7b^2 \neq 0\) for all \(a, b \in \mathbb{Q}\) where \(a\) and \(b\) are not both 0. [2]
  2. A set \(G\) is defined by \(G = \{a + b\sqrt{7} : a, b \in \mathbb{Q}, a\) and \(b\) not both 0\(\}\). Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.) [7]
  3. A subset \(H\) of \(G\) is defined by \(H = \{1 + c\sqrt{7} : c \in \mathbb{Q}\}\). Determine whether or not \(H\) is a subgroup of \((G, \times)\). [2]
  4. Using \((G, \times)\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false. [2]
OCR MEI Further Extra Pure Specimen Q2
4 marks Challenging +1.2
A binary operation \(*\) is defined on the set \(S = \{p, q, r, s, t\}\) by the following composition table.
\(*\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(q\)\(q\)\(p\)\(s\)\(t\)\(r\)
\(r\)\(r\)\(t\)\(p\)\(q\)\(s\)
\(s\)\(s\)\(r\)\(t\)\(p\)\(q\)
\(t\)\(t\)\(s\)\(q\)\(r\)\(p\)
Determine whether \((S, *)\) is a group. [4]