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GroupsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

  • State the four group axioms.
  • What is the order of a group?
  • What is the order (period) of an element a?
  • What is a subgroup?
  • Trivial subgroup?
  • Proper subgroup?
  • What does a Cayley table of a group show about each row and column?
  • What is a cyclic group?
  • What is an abelian group?
  • Does cyclic imply abelian? Does abelian imply cyclic?
  • Order of the symmetry group of a regular n-gon?
  • Why is (\mathbb{Z},\times) not a group?

Exam questions on Groups

  1. The set G={1,5,7,11}G=\{1,5,7,11\} under multiplication modulo 12.
    Determine whether GG is cyclic, justifying your answer.2 marks
  2. The set TT of symmetries of an equilateral triangle forms a group under composition: the identity, rotations through 120∘120^\circ and 240∘240^\circ about the centre, and reflections in the three lines of symmetry.
    Explain why TT is not abelian.2 marks
  3. The set G={0,1,2,3,4,5}G=\{0,1,2,3,4,5\} under addition modulo 6, written +6+_6.
    Given that GG is closed under +6+_6, show that (G,+6)(G,+_6) satisfies the other three group axioms.3 marks
See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).