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 $A_4$ was pretty

straigthforward and I'm half way through proving that the

_octahedral group_ is just $S_4$ (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 $A_5$

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 $S_6$ (using the main

diagonals on the $12$ vertices of the icosahedron) and identifying this

subgroup as $A_5$. 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

$72^o$ counter-clockwise in the top face we get the permutation in

$S_{20}$

$(1,2,3,4,5)(6,8,10,12,14)(7,9,11,13,15)(16,17,18,19,20)$

and the

rotation by $72^o$ 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 $S_{20}$ 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 $A_5$. 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…

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