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 S_{5} 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?

Can you use

Can you give an algorithm for using

Can you prove that

Can you find

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

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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