Permutation puzzles

Apply the permutations by clicking the buttons. line from x to f(x). This button only has an effect when f is not the identity. f is the permutation function created by composition of a and b according to order in which they are clicked. These lines can be shown to more clearly show where a dot has ended up compared to where it starts, though this can also be seen from the numbers and colours of the dots.

What subgroup of S5 do a and b generate?
Can you use a and b to switch dots 1 and 2, while returning all other dots to their original positions? If you can, how? If not, why not? 3 2 1 5 4 3 2 1 5 4

Can you give an algorithm for using a and b to switch any two dots, while returning all other dots to their original positions?
Can you prove that a and b generate all of S12?

a and b generate the dihedral group D12. Verify that ab=ba-1. The group presentation is ⟨ a, b : a12=b2=baba=1 ⟩. Verify that all reflections in D12 can be obtained by conjugation of b. This groups acts on the same set X={1,2,3,4,5,6,7,8,9,10,11,12} as in the previous example.

Can you find b in terms of a?

Now X={1,2,3,4,5,6,7,8}, and the group is generated by an element of order 8 and an element of order 2. What is it? Hint: can you find bab in terms of a? This is not a particularly obvious group, but you should be able to locate it here.

Cayley's theorem tells us that all groups are subgroups of permutation groups. So, if you understand everything about permutation groups, you understand everything about groups! These examples give a little bit of practice. I recommend GAP for experimenting with groups. Download, install and run, then you can type in things like: Size(Group((1,2,3,4,5,6,7,8),(1,5))); to find the number of elements in the permutation group generated by (1,2,3,4,5,6,7,8) and (1,5). Also the Wikipedia page on the symmetric group and Groups of order 16 might be useful.

Some nice group theory pages (none of these due to me; happy to add more!): Lots of group multiplication tables; Group names; Wiki group properties.
Mathematics of the Rubik's cube (MIT notes); the classic for the mathematics of the Rubic's cube is Singmaster's work.
I made an html verision of one of my algebra quizzes, but right now I'm making them all in VeVox, and don't have enough time to duplicate anything to share.

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