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 -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own