Let’s begin by recalling a particularly simple result about certain finite groups: every finite group of prime order is cyclic. To see this, suppose is a finite group of order , and choose an element of . The group generated by is a subgroup of . By Lagrange, we know that divides ; but and is prime, so . Since , we observe that , i.e., is cyclic. It’s also possible to ask for a classification of finite groups of, say, order (the only groups are and the Klein 4-group , which isn’t cyclic). Now, one might ask for a general classification of finite groups.
This seems like a ridiculously hard problem, and that’s because it is. One very important result helping to (somewhat) simplify this problem is the Jordan-Hölder theorem. A simple group is a group whose normal subgroups are only the trivial group or the whole group. Let be a group; a composition series for is a sequence of subgroups of such that the quotient (called a composition factor) is simple, i.e., such that is a maximal strict normal subgroup of . The Jordan-Hölder theorem is as follows: The lengths of any two composition series of a group are the same, and their composition factors are isomorphic. This is particularly applicable in the case when is finite, because in this case, you can say that is “built up” from (automatically finite) simple groups; so we can reduce our classification of finite groups to a classification of the finite simple groups, which should (hopefully) be simpler.
Now, I won’t tell you how you can classify all finite simple groups – that’s way too long, complicated, and much beyond my knowledge. (A history is given at Wikipedia.) Anyway, it turns out that the finite simple groups are classified (this is highly nontrivial; it took about 500 journal articles to prove this!) into the following classes of groups:
- Cyclic groups of prime order.
- Alternating groups of degree at least 5.
- A “finite simple group of Lie type over a finite field” (there are 16 families of such groups).
- 26 weird groups.
This last class of groups is what interests us – these are the sporadic groups. Here’s a way of stating this last point in the statement above:
- There are unique simple groups of the following orders, which are not in any of the families described above: 7920, 95040, 443520, 10200960, 244823040, 175560, 604800, 50232960, 86775571046077562880, 4157776806543360000, 42305421312000, 495766656000, 64561751654400, 4089470473293004800, 1255205709190661721292800, 44352000, 898128000, 4030387200, 145926144000, 448345497600, 460815505920, 273030912000000, 51765179004000000, 90745943887872000, 4154781481226426191177580544000000, 8080174247945128758and 86459904961710757005754368000000000
These are denoted by , , , , (called the Mathieu groups), , , , (called the Janko groups), , , (called the Conway groups), , , (called the Fischer groups), (called the Higman-Simms group), (called the McLaughlin group), (called the Held group), (called the Rudvalis group), (called the Suzuki group), (called the O’Nan group), (called the Harada-Norton group), (called the Lyons group), (called the Thompson group), (the baby monster), and (called the Monster group). I’d like to talk about how (some of) these groups relate to something called the Leech lattice in a future post; for now, I just want to mention a really cool observation, called monstrous moonshine, which concerns one of these sporadic groups.
Recall that any elliptic curve can be identified with a two-dimensional lattice of , and the Weierstrass -function gives a bijection between and an elliptic curve in . Similarly, the -invariant gives a bijection between and , i.e., . This function is an incredibly important one in number theory; for example, the modular functions of weight zero for are the rational functions of . The -expansion of the -invariant is given by: , where . I won’t go through the whole definition, because that’s not as relevant as the coefficients of this function.
Let’s make a neat little algebraic observation. The dimensions of the irreducible representations of the monster group are (sequence A001379 in OEIS): 1, 196883, 21296876, 842609326, 18538750076, … Denoting these (as wikipedia does) by , respectively, we observe that , , , , , etc. So there’s a deep connection between the monster and the j-function; this is called monstrous moonshine (by Conway and Norton). I’ll refer the reader to this paper, which does justice to this awesome observation. This relationship between the monster and the j-function has been an interesting source of mathematics – for example, is this limited only to the monster? Could there be such a relationship for general sporadic groups? (I’m nowhere near an expert at this stuff, so I won’t say more at risk of saying something incorrect.) In a future post, I’ll try to talk about the Leech lattice.