In my geometry 101 course I'm doing the rotation-symmetry groups
of the Platonic solids right now. This goes slightly slower than
expected as it turned out that some secondary schools no longer give a
formal definition of what a group is. So, a lot of time is taken up
explaining permutations and their properties as I want to view the
Platonic groups as subgroups of the permutation groups on the vertices.
To prove that the _tetrahedral group_ is isomorphic to A4 was pretty
straigthforward and I'm half way through proving that the
_octahedral group_ is just S4 (using the duality of the octahedron
with the cube and using the 4 body diagonals of the cube).
Next
week I have to show that the _icosahedral group_ is isomorphic to A5
which is a lot harder. The usual proof (that is, using the duality
between the icosahedron and the dodecahedron and using the 5 cubes
contained in the dodecahedron, one for each of the diagonals of a face)
involves too much calculations to do in one hour. An alternative road is
to view the icosahedral group as a subgroup of S6 (using the main
diagonals on the 12 vertices of the icosahedron) and identifying this
subgroup as A5. A neat exposition of this approach is given by John Baez in his
post Some thoughts on
the number 6. (He also has another post on the icosahedral group
in his Week 79's
finds in mathematical physics).
But
probably I'll go for an “In Gap we
thrust”-argument. Using the numbers on the left, the rotation by
72o counter-clockwise in the top face we get the permutation in
S20
(1,2,3,4,5)(6,8,10,12,14)(7,9,11,13,15)(16,17,18,19,20)
and the
rotation by 72o counterclockwise along the face (1,2,8,7,8) gives
the permutation
(1,6,7,8,2)(3,5,15,16,9)(4,14,20,17,10)(12,13,19,18,11)
GAP
calculates that the subgroup dode of S20 generated by these two
elements is 60 (the correct number) and with IsSimplegroup(dode); we
find that this group must be simple. Finally using
IsomorphismTypeInfoFiniteSimplegroup(dode);
we get the required
result that the group is indeed isomorphic to A5. The time saved I
can then use to tell something about the classification project of
finite simple groups which might be more inspiring than tedious
calculations…