From: torquemada@my-dejanews.com
Subject: Re: Monster Group and The Happy Family
Date: Fri, 12 Mar 1999 23:32:44 GMT
Newsgroups: sci.math
Keywords: What are the sporadic simple groups?
In article <36E8ED96.C3F98731@rdg.ac.uk>,
Kevin Anderson wrote:
> I've found a book in the library called "Twelve Sporadic Groups". Now I
> only know the basics of group theory, but I was intrigued by the phrases
> "the monster group" and "the happy family" found therein. Could someone
> provide a rough explanation of what they are and how they are
> constucted?
>
> Thanks for any reply.
I guess you know what a homomorphism is. A simple group is a group G such that
if you have a homomorphism f:G->H where H is any other group then the kernel
of f is trivial (or equivalently the image of f is isomorphic to g).
All of the simple groups have been classified. Mostly they form series: for
example the cyclic groups C_p of prime order p or the alternating groups
A_n starting with the famous n=5 case.
However it turns out that there are precisely 26 left over 'sporadic' groups
that don't really fit into any series. The largest is called the monster
group. It is one hell of an amazing group. For example the dimensions
of the linear representations of this group crop up (slightly disguised)
in the power series of the modular j-function that appears in classical
complex function theory. This is amazing because these subjects have no
obvious relationship to each other. At first the numerical coincidence was
unexplained and given the name "The Monstrous Moonshine Conjectures" and
recently they were proved by Fields medalist Richard Borcherds - although
the proof is just a mechanical proof that yields little insight. And the
really weird thing is that the proof borrows material from, of all
places, String Theory.
The Monster is also connected to the Leech Lattice - a miraculous
lattice in 24 dimensions that has among its myriad properties the
claim of being the densest known sphere packing in 24D. (The details of the
construction are quite hard though.) One of the sporadic groups, Co_1,
is in fact the automorphism group of this lattice modulo a group of order 2.
Similarly the other 'Conway' groups Co_2 and Co_3 come from the
automorphism group.
Many of the other sporadic groups are related to the Monster Group in various
ways - as well as being interesting in their own right. For example
there are the Mathieu groups M_12 and M_24 which are related to the
Golay code (a rare 'perfect' error correcting code), the 'Rubik icosahedron',
the game played with coins called 'Mogul' and the experimental design known as
S(5,8,24) (which in turn is related to the Leech lattice).
Anyway...what I say might not have made any sense but you now have enough
keywords to do a good web search! There isn't enough space even to
scratch the surface!
--
Torque
http://www.tanelorn.demon.co.uk
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