How do we compute the order of the Monster group?

How do we compute the order of the Monster group?

The answer is quoted in many places, but when I trace back the references, I can't find any place where it's computed, or even a sketch of the computation. They mostly point back to some article by Griess which is listed as "to appear" or "in preparation" but I can't find anywhere the article actually appeared.

For example, in Griess' article "The Friendly Giant", he writes in Section 15 (p. 96):

Many properties of this hypothetical simple group [this refers to the Monster group] were derived, including ... a correct guess of its order, using the result of Frobenius which says that the cardinality of $\{g \in G: g^n = 1\}$ is divisible by $n$, for any finite group $G$ and integer dividing $|G|$ (a proof that its order is the number of Sect. 1 was written down by Griess [36]).

[36] is listed as "The Structure of the Friendly Giant", in preparation. It is not clear to me how one would use that theorem to compute, or even to guess, the order of the Monster.

Elsewhere in the same article, in Lemma 2.16 (p. 11), Griess quotes a theorem of Steve Smith (from a paper entitled "Large extraspecial groups of width 4 and 6") saying that the order of a hypothetical group satisfying certain conditions is the order of the Monster. But Smith's article cites another article of Griess "The Structure of the Monster Simple Group" (which could be the article referenced above as [36] under a changed name, but in any case it doesn't contain the calculation mentioned), and that article cites another article of Griess listed as "to appear": "On the subgroup structure of the group of order $2^{46} 3^{20} \ldots$" [the actual citation gives the full prime factorization of the Monster order]. I haven't been able to find that article, if it has appeared.

Wikipedia mentions that the Thompson order formula is used (and a similar comment is made on p. 183 of the non-technical account "Symmetry and the Monster" by Mark Ronan), without giving more details. This seems plausible because the order of the centralizers of the two involution classes were known, but it is not clear how one would compute the other terms in the Thompson order formula.

Another article by Griess is "Schur multipliers of some sporadic simple groups" where in the introduction (p. 446), Griess says that there is "strong evidence that a simple group ... of order $2^{46} 3^{20} \dots$ exists" [again, the full factorization is given in the article]. But this is followed by 3 citations which are not accessible: a lecture by Fischer in 1973, the "to appear" article above "On the subgroup structure of...", and some unpublished work by Thompson.

So how do we compute the order the Monster? Even if we don't know exactly how it was originally done, how would we do it now? I'm looking for even a sketch of a proof of a theorem that says if a group satisfies some simple conditions that force it to be the Monster (like having two involution classes of the known centralizers, and maybe some other conditions that are needed), then its order is the order of the Monster.

Is it too much to expect that one could compute the order directly by some counting argument using the construction by Griess, Conway, or any subsequent construction?

Solution 1:

After more searching, I found a satisfactory answer in the paper "A uniqueness proof for the Monster" by Griess, Meierfrankenfeld, and Segev. The main theorem states:

Let $G$ be a finite group containing two involutions $a$ and $z$ such that $C_G(a)$ is of $2 \cdot F_2$-type and $C_G(z)$ is of $2_+^{1+24}.{\rm Co}_1$-type. Then $G$ is unique up to isomorphism.

And then Corollary 3.7.3 gives the order $|G|$, which is computed by summing the numbers in the rightmost column of Table VII (3.4.3), plus the numbers in the rightmost column of Table IX (3.4.8.1), plus 1, and then multiplying this total sum by $|2 \cdot F_2|$.