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