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Author: lievenlb

simple groups

I
found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front
cover the list of the 26 _known_ sporadic groups together with a
starred added in proof saying

  • added in
    proof … the classification of finite simple groups is complete.
    there are no other sporadic groups.

(click on the left picture to see a larger scanned image). In it is a
beautiful paper by John Conway “Monsters and moonshine” on the
classification project. Along the way he describes the simplest
non-trivial simple group $A_5 $ as the icosahedral group. as well as
other interpretations as Lie groups over finite fields. He also gives a
nice introduction to representation theory and the properties of the
character table allowing to reconstruct $A_5 $ only knowing that there
must be a simple group of order 60.
A more technical account
of the classification project (sketching the main steps in precise
formulations) can be found online in the paper by Ron Solomon On finite simple
groups and their classification
. In addition to the posts by John Baez mentioned
in this
post
he has a few more columns on Platonic solids and their relation to Lie
algebras
, continued here.

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sanglier

Although we can hear every night (in France) wild
boar
descending the mountain in search for water and though
shooting them seems to be the only (male) social activity in Sablieres
this time of year, it is much easier to prepare a wild boar stew in
Belgium than in the Ardeche-
Cevenole
. The reason being that it is impossible to find wild boar
meat in a supermarket or butcher in the region. A few years ago we
went there in the winter and I wanted to prepare wild boar for
new-year's eve but every butcher we approached for meat replied
with a fairly angry 'ne fais pas!' as if we asked him to
perform some illegal act (which probably, unknowingly, we did…).
In contrast, supermarkets and butchers are flooded with duck
('fillet de canard', 'cuisse de canard', 'confit
de canard', you name it) but I've never spotted a living duck
in the entire region! It turned out that, in order to prevent people
from shooting boar to make a living, it is not allowed to sell wild
boar meat. The only possibility to get it is either to shoot one
yourself or to have a friend in the local hunting society (les
chasseurs, omnipresent and rather political influencial in France).
Once you managed the latter, you have to master the basics of the
exchange-trade (you always have to give one commodity to get another).
The first time we offered a bottle of Pastis (Ricard) which was
flatly rejected because to them the value of Pastis is much higher
than that of boar meat, but they would accept a bottle of wine
instead…
Anyway, here is a fairly erratic recipe for wild
boar (serves 4 to 6) as I made it on thursday : get 1kg to 1.5kg of
wild boar meat, defrost if necessary and slice it into 1-2-3cm
cubicles (don't take this too strict). Meanwhile, heat up plenty
of olive oil, slice up two fairly large onions and as much garlic as
you (and your dinner partners) can master and fry this in the oil. Add
the sliced up meat and let it fry until all pieces are nicely brown.
Add pepper and your favourite herb (which is Thyme in my case, but
Oregano or Rosemary might be other choices). Add half of bottle (or
more) of red wine and some water (change percentages if you have
younger children) and heat it up. Have a look in the refrigerator for
vegatables that would go well with the meat. This time I used carrots,
mushroom and courgettes as they were available but use your own taste
to change this. Slice up 6 big carrots and add them (in general :
slice up the harder vegetables and add them now, keep the softer
vegetables for later) and let all of this stew for 2 hrs or more on a
moderate fire (stir every 10 minutes and add wine/water when needed).
Half an hour before serving add the courgettes (or any other softer
vegetable) and 15minutes later the mushrooms. Should be accompanied
with a proper 'gratin' but as I cannot approximate the one
made by Michel, chef of camping la drobie and part-time mayor of the local
village, I went for Rosti on thursday. Enjoy!

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icosahedral group

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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