# How was the Monster's existence originally suspected?

I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.

For example, in ncatlab's article,

The Monster group was predicted to exist by Bernd Fischer and Robert Griess in 1973, as a simple group containing the Fischer groups and some other sporadic simple groups as subquotients. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of $M$ and discovered other properties and subgroups, assuming that it existed.

Or Wikipedia,

The Monster was predicted by Bernd Fischer (unpublished) and Robert Griess (1976) in about 1973 as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months the order of $M$ was found by Griess using the Thompson order formula [...]

Its existence is a non-trivial fact: when the original moonshine conjectures were made, mathematicians suspected its existence, and had been able to work out its character table, but could not prove it actually existed. They did know that the dimensions of the smallest irreducible representations would be 1, 196883; and 21296876.

It surprises me that this object could have been predicted before being rigorously discovered, due to it being often described as very complicated and highly nonobvious (or at least its construction).

Take for instance the description in this AMS review of Moonshine Beyond the Monster:

The proof of the moonshine conjectures depends on several coincidences. Even the existence of the monster seems to be a ﬂuke in any of the known constructions: these all depend on long, strange calculations that just happen to work for no obvious reason, and would not have been done if the monster had not already been suspected to exist.

It'd be cool to be acquainted with this part of the story in some more detail at an accessible level, if possible, though I realize it may necessarily involve heavy machinery or convoluted calculations.

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My (very) vague impression is this: you work on the classification of finite simple groups, and there is one case which you can't quite rule out. You can prove a lot of stuff about it, and none of it seems to lead to a contradiction, so maybe this case actually occurs... –  Qiaochu Yuan Aug 20 '12 at 2:40
+1 I think this is one of the best questions I've seen of late. –  Alex Becker Aug 20 '12 at 2:45
I can recommend Mark Ronan, Symmetry and the Monster, as a very good work of high-level exposition on the topic. –  Gerry Myerson Aug 20 '12 at 3:38
@GerryMyerson Thanks. // I also found a lecture by the same title and author/speaker on vimeo. –  blue Aug 25 '12 at 15:05

I'm not an expert on this topic, and have only read summaries of the proofs and techniques. Nonetheless, let me give this a try.

By the odd order theorem, any non-abelian simple group has a nontrivial involution. One of the key techniques in the classification of finite simple groups was to study the centralizers of involutions. That is, you try to find all finite simple groups such that the centralizer of an involution has a specific form.

Once you prove enough results you end up in the following situation. You look at candidate centralizers of involutions which are made up entirely of the simple groups that you already know. You use this to try to find candidates for any new simple groups that you didn't know already. If your search turns up a new example, you repeat the process trying to find new candidate centralizers of involution involving the new simple group.

Thus it should not be surprising that when people found the Baby Monster they started looking for candidates centralizers of involutions which you can make out of the Baby Monster. Once you find a good candidate that you can't rule out (e.g. they couldn't find any reason why the centralizer of an involution couldn't be the double cover of the Baby Monster), then you conjecture a new simple group.

What's interesting is that when you get to the Monster you can try repeating this process to try to find yet another new simple group where the centralizer of an involution is built from the Monster. There's no reason a priori that this couldn't continue forever, with more and more sporadic groups. But it turns out that it stops at the Monster.

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In the second paragraph by "simple group" you mean "non-abelian finite simple group," right? –  Qiaochu Yuan Aug 20 '12 at 3:35
Yup, of course, fixed. Thanks! –  Noah Snyder Aug 20 '12 at 3:36
From a historical point of view, it was proved in a 1955 Annals paper of R. Brauer and K.A. Fowler that given an involution $t$ in a finite non-Abelian simple group $G,$ then $|G|$ is bounded (in an explicit way) in terms of $|C_{G}(t)|$. So this paper was written before the Feit-Thompson odd order theorem was proved. But, as you (Noah) say, the Feit-Thompson theorem and the Brauer-Fowler theorem suggested a programme for classifying the finite simple groups which was ultimately successful. Several sporadic simple groups were discovered via centralizer of involution characterizations. –  Geoff Robinson Aug 20 '12 at 5:01
There was a really nice thread on Mathoverflow which said something along these lines. Can anyone find (and then post!) the link? I can't seem to find it... –  user1729 Aug 20 '12 at 9:16