Download - Group theory in the classroom
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Group theory in the classroom
Danny Brown
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outline
• What is a group?• Symmetry groups• Some more groups• Permutations• Shuffles and bell-ringing• Even more symmetry
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Rotation and reflection
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Direct and indirect symmetries
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what is a group?
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Some other groups
• Symmetries of other shapes• Clock arithmetic• Matrices• Complex numbers• ...
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Groups of order 4
How many are there? Discuss.
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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…
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Breaking symmetry
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Permutations
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Permutations
1 22 3
3 1
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Permutations
1 22 3
3 1= ( 1 2 3 )
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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 )
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Transpositions
In fact, all permutations can be expressed as product of transpositions…
What does this mean geometrically?
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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.
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Shuffle factory (NRICH)
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Perfect shuffles
• ‘Monge’ shuffle• ‘Riffle’ shuffle• ‘Two-pile’ shuffle• …
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Riffle shuffle LEFT
RIGHT
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Monge shuffle 1 2
3 4
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Monge shuffle permutation
1 3
2 2
3 4
4 1= ( 1 3 4 ) ( 2 )
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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?
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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 about
the direct and indirect symmetries?
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Can you invent a perfect shuffle of your own?