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.
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;
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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