Finite simple groups, sporadic groups, and the monster

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 G is a finite group of order p, and choose an element g\neq 0 of G. The group \langle g\rangle generated by g is a subgroup of G. By Lagrange, we know that |\langle g\rangle| divides |G|; but g\neq 0 and |G|=p is prime, so |\langle g\rangle|=p. Since \langle g\rangle\leq G, we observe that \langle g\rangle\cong G, i.e., G is cyclic. It’s also possible to ask for a classification of finite groups of, say, order 4 (the only groups are \mathbf{Z}/4\mathbf{Z} and the Klein 4-group \mathbf{Z}/2\mathbf{Z}\times\mathbf{Z}/2\mathbf{Z}, 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 G be a group; a composition series for G is a sequence 1\triangleleft \cdots H_i\triangleleft H_{i+1}\triangleleft H_n = G of subgroups of G such that the quotient H_{i+1}/H_i (called a composition factor) is simple, i.e., such that H_i is a maximal strict normal subgroup of H_{i+1}. 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 G is finite, because in this case, you can say that G 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:

  1. Cyclic groups of prime order.
  2. Alternating groups of degree at least 5.
  3. A “finite simple group of Lie type over a finite field” (there are 16 families of such groups).
  4. 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 M_{11}, M_{12}, M_{22}, M_{23}, M_{24} (called the Mathieu groups), J_1, J_2, J_3, J_4 (called the Janko groups), Co_1, Co_2, Co_3 (called the Conway groups), Fi_{22}, Fi_{23}, F_{3+} (called the Fischer groups), HS (called the Higman-Simms group), McL (called the McLaughlin group), F_7 (called the Held group), Ru (called the Rudvalis group), Suz (called the Suzuki group), O'N (called the O’Nan group), HN (called the Harada-Norton group), Ly (called the Lyons group), Th (called the Thompson group), B (the baby monster), and M (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 L of \mathbf{C}, and the Weierstrass \wp-function gives a bijection between \mathbf{C}/L and an elliptic curve in \mathbf{CP}^2. Similarly, the j-invariant gives a bijection between \mathrm{SL}_2(\mathbf{Z})/(\mathfrak{H}\cup\{\infty\}\cup\mathbf{Q}) and \mathbf{CP}^1, i.e., S^2. This function is an incredibly important one in number theory; for example, the modular functions of weight zero for \mathrm{SL}_2(\mathbf{Z}) are the rational functions of j. The q-expansion of the j-invariant is given by: j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\dots, where q = e^{2 \pi i \tau}. 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 r_n, respectively, we observe that 1 = r_1, 196884 = r_1 + r_2, 21493760 = r_1 + r_2 + r_3, 864299970 = 2r_1 + 2r_2 + r_3 + r_4, 20245856256 = 3r_1 + 3r_2 + r_3 + 2r_4+ r_5, 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.

Have fun,
SKD

 

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