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• Clock arithmetic• Matrices• Complex numbers• ...

Groups of order 4

How many are there? Discuss.

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…

Breaking symmetry

Permutations

Permutations

1 22 3

3 1

Permutations

1 22 3

3 1= ( 1 2 3 )

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 )

Transpositions

In fact, all permutations can be expressed as product of transpositions…

What does this mean geometrically?

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.

Shuffle factory (NRICH)

Perfect shuffles

• ‘Monge’ shuffle• ‘Riffle’ shuffle• ‘Two-pile’ shuffle• …

Riffle shuffle LEFT

RIGHT

Monge shuffle 1 2

3 4

Monge shuffle permutation

1 3

2 2

3 4

4 1= ( 1 3 4 ) ( 2 )

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?

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?

Can you invent a perfect shuffle of your own?