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Isomorphism of groupsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

  • Definition: isomorphism \phi:G\to H?
  • What does G\cong H mean?
  • What does an isomorphism do to the identity?
  • What does an isomorphism do to the order of an element?
  • Must isomorphic groups have the same order?
  • Two cyclic groups of the same order: are they isomorphic?
  • How can you show two groups are not isomorphic?
  • Is \{1,3,5,7\} modulo 8 isomorphic to C4?
  • Is the symmetry group of an equilateral triangle isomorphic to C6?
  • Isomorphism from \{0,1,2,3\} under +4 to \{1,i,-1,-i\}?
  • Generator of \{1,\dots,6\} under multiplication modulo 7?
  • Do isomorphic groups need the same operation or the same elements?

Exam questions on Isomorphism of groups

  1. The set G={1,i,−1,−i}G=\{1,i,-1,-i\} under complex multiplication, and the set H={0,1,2,3}H=\{0,1,2,3\} under addition modulo 4. The function ϕ:H→G\phi:H\to G is defined by ϕ(n)=in\phi(n)=i^n.
    Verify that ϕ(2+43)=ϕ(2)×ϕ(3)\phi(2+_43)=\phi(2)\times\phi(3).2 marks
  2. The set G={1,3,5,7}G=\{1,3,5,7\} under multiplication modulo 8, and the cyclic group C4={e,a,a2,a3}C_4=\{e,a,a^2,a^3\} with a4=ea^4=e.
    Deduce that GG and C4C_4 are not isomorphic.2 marks
  3. The set A={0,1,2,3,4,5}A=\{0,1,2,3,4,5\} under addition modulo 6, and the set B={1,2,3,4,5,6}B=\{1,2,3,4,5,6\} under multiplication modulo 7.
    Show that 33 is a generator of BB.3 marks
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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).