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IsomorphismEdexcel A-Level Further Maths: Flashcards

What these 12 flashcards ask

  • Define an isomorphism between groups G and H.
  • What does a bijection mean?
  • What does an isomorphism do to the identity?
  • What does an isomorphism do to inverses?
  • What do isomorphisms preserve about elements?
  • Name four properties preserved by isomorphism.
  • Why are Z4 and \{1,3,5,7\} mod 8 not isomorphic?
  • Why are Z6 and S3 not isomorphic?
  • How do you prove G cyclic of order n is isomorphic to Zn?
  • Isomorphism from \{1,i,-1,-i\} to Z4?
  • How many structures of group of prime order p?
  • Is it enough to find two groups of the same order to claim they are isomorphic?

Exam questions on Isomorphism

  1. Let G={1,i,−1,−i}G=\{1,i,-1,-i\} under multiplication of complex numbers and H={0,1,2,3}H=\{0,1,2,3\} under addition modulo 4. Both are groups of order 44.
    Explain why no isomorphism from GG to HH can map ii to 22.2 marks
  2. Let K={1,3,5,7}K=\{1,3,5,7\} under multiplication modulo 8 and Z4={0,1,2,3}Z_4=\{0,1,2,3\} under addition modulo 4. Both are groups of order 44.
    Give a second reason, based on whether the groups are cyclic, why KK and Z4Z_4 are not isomorphic.2 marks
  3. Let G={0,1,2,3,4,5}G=\{0,1,2,3,4,5\} under addition modulo 6 and H={1,2,3,4,5,6}H=\{1,2,3,4,5,6\} under multiplication modulo 7, both of which are groups. Define φ:G→H\varphi:G\to H by φ(k)=3k mod 7\varphi(k)=3^{k}\bmod 7.
    Find φ(k)\varphi(k) for each k∈Gk\in G, and hence show that φ\varphi is a bijection.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).