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Group theory in the classroom

Group theory in the classroom. Danny Brown. outline. What is a group? Symmetry groups Some more groups Permutations Shuffles and bell-ringing Even more symmetry. Rotation and reflection. Direct and indirect symmetries. what is a group?. Some other groups. Symmetries of other shapes

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Group theory in the classroom

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  1. Group theory in the classroom Danny Brown

  2. outline • What is a group? • Symmetry groups • Some more groups • Permutations • Shuffles and bell-ringing • Even more symmetry

  3. Rotation and reflection

  4. Direct and indirect symmetries

  5. what is a group?

  6. Some other groups • Symmetries of other shapes • Clock arithmetic • Matrices • Complex numbers • ...

  7. Groups of order 4 How many are there? Discuss.

  8. Fermat’s little theorem For odd primes p, with a and p co-prime. ‘Proof’: Multiplication on the integers {1,2,…,p-1} is a group. … and for any element g in a group of size n…

  9. Breaking symmetry

  10. Permutations

  11. Permutations 1  2 2  3 3  1

  12. Permutations 1  2 2  3 3  1 = ( 1 2 3 )

  13. Transpositions We know ( 1 2 3 ) is the same as P …which is the same as T then S …which is the same as ( 1 2 ) then ( 1 3 )

  14. Transpositions In fact, all permutations can be expressed as product of transpositions… What does this mean geometrically?

  15. Bell-ringing / braids Q. Can you cycle through all the permutations of 1 2 3 using just one transposition? Explain why. Q. Can you cycle through all the permutations of 1 2 3 using just two transposition? Explain why.

  16. Shuffle factory (NRICH)

  17. Perfect shuffles • ‘Monge’ shuffle • ‘Riffle’ shuffle • ‘Two-pile’ shuffle • …

  18. Riffle shuffle LEFT RIGHT

  19. Monge shuffle 1 2 3 4

  20. Monge shuffle permutation 1  3 2  2 3  4 4  1 = ( 1 3 4 ) ( 2 )

  21. Shuffles and symmetry We have seen that permutations of 1 2 3 are symmetries of the triangle…. …but how can we describe the Monge shuffle (1 3 4) (2) geometrically? …what about the ‘riffle’ shuffle? How about permutations of 4 numbers generally?

  22. Permutations of 4 elements • How many are there? • Can you find them all? • Are there any patterns? • What symmetries do they represent? • What do you notice aboutthe direct and indirect symmetries?

  23. Can you invent a perfect shuffle of your own?

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