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I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them and would like some intuition about some things.

For example, the order of the monster group is over $8\times 10^{53}$, yet it is simple, so it has no normal subgroups...how? What is so special about the prime factorization of its order? Why is it $2^{46}$ and not $2^{47}$? Why is it not possible to extend it to obtain that additional power of 2 without creating a normal subgroup? Some of the properties seem really arbitrary, and yet must be very fundamental to the algebra of groups.

I don't think I'm the only person curious about this, but I hesitated posting due to my relative inexperience.

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One might even ask, why are there sporadic simple groups? If you are looking for intuition, maybe you should add the [soft-question] tag and make it wiki? –  Sonia Balagopalan Mar 9 '10 at 16:34
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I feel like being perverse and remarking that all but finitely many simple groups are larger than any sporadic group, so the sporadic groups are actually comparatively small... I've a feeling the answer is going to be a disappointing something along the lines of how the smaller groups don't have enough room in which to be sporadic. –  some guy on the street Mar 9 '10 at 16:51
    
the other thing that occurs to me is that the sporadics I can sort-of remember are built as automorphism groups of certain combinatorial structures; now, those structures themselves look like generalizing in a few ways, but doing so could lead to several possibilities: e.g. their symmetries might split too much, so that the simple subgroups are too small; the desired generalization might not always exist; or the symmetry groups of the new structures may for arcane reasons be enumerated already among the classical groups. –  some guy on the street Mar 9 '10 at 17:00
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As a general comment, there are all sorts of "low-dimensional" obstructions restricting the isomorphism types of groups of small order, such as those coming from the Sylow theorems, and these obstructions become vanishingly less likely as you look at groups of large order. –  Qiaochu Yuan Mar 9 '10 at 17:22

5 Answers 5

up vote 51 down vote accepted

The question seems to be made of several smaller questions, so I'm afraid my answer may not seem entirely coherent.

I have to agree with the other posters who say that the sporadic simple groups are not really so large. For example, we humans can write down the full decimal expansions of their orders, where a priori one might think we'd have to resort to crude upper bounds using highly recursive functions. (In contrast, one could say that almost of the groups in the infinite families are too large for their orders to have a computable description that fits in the universe.) Furthermore, as of 2002 we can load matrix representatives of elements into a computer, even for the monster. Noah pointed out that the monster has a smaller order than $A_{50}$, but I think a more apt comparison is that the monster has a smaller order than even the smallest member of the infinite $E_8$ family. Of course, one could ask why $E_8$ has dimension as large as 248...

There was a more explicit question: how is it possible that a group with as many as $8 \times 10^{ 53 }$ elements doesn't have any normal subgroups? I think the answer is that the order of magnitude of a group says very little about its complexity. There are prime numbers very close to the order of the monster, and there are simple cyclic groups of those orders, so you might ask yourself why that fact doesn't seem as conceptually disturbing. Perhaps slightly more challenging is the fact that there aren't any elements of order greater than 119, but again, there is work on the bounded and restricted Burnside problems that shows that you can have groups of very small exponent that are extremely complicated.

A second point regarding the large lower bound on order is that there are smaller groups that could be called sporadic, in the sense that they fit into reasonably natural (finite) combinatorial families together with the sporadics, but they aren't designated as sporadic because small-order isomorphisms get in the way. For example, the Mathieu group $M_{10}$ is the symmetry group of a certain Steiner system, much like the simple Mathieu groups, and it is an index 11 subgroup of $M_{11}$. While it isn't simple, it contains $A_6$ as an index 2 subgroup, and no one calls $A_6$ sporadic. Similarly, we describe the 20 "happy family" sporadic subquotients of the monster, but we forget about the subquotients like $A_5$, $L_2(11)$, and so on. Since the order of a nonabelian simple group is bounded below by 60, there isn't much room to maneuver before you get to 7920, a.k.a. "huge" range.

The question about the why the 2-Sylow subgroup has a certain size is rather subtle, and I think a good explanation would require delving into the structure of the classification theorem. A short answer is that centralizers of order 2 elements played a pivotal role in the classification after the Odd Order Theorem, and there was a separation into cases by structural features of centralizers. One of the cases involved a centralizer that ended up having the form $2^{1 + 24} . Co1$, which has a 2-Sylow subgroup of order $2^{46}$ (and naturally acts on a double cover of the Leech lattice). This is the case that corresponds to the monster.

Regarding the prime factorization of the order of the monster, the primes that appear are exactly the supersingular primes, and this falls into the general realm of "monstrous moonshine". I wrote a longer description of the phenomenon in reply to Ilya's question, but the question of a general conceptual explanation is still open.

I'll mention some folklore about the organization of the sporadics. There seems to be a hierarchy given by

  • level 0: subquotients of $M_{24}$ = symmetries of the Golay code
  • level 1: subquotients of $Co1$ = symmetries of the Leech lattice, mod $\{ \pm 1 \}$
  • level 2: subquotients of the monster = conformal symmetries of the monster vertex algebra

where the groups in each level naturally act on (objects similar to) the exceptional object on the right. I don't know what explanatory significance the sequence [codes, lattices, vertex algebras] has, but there are some level-raising constructions that flesh out the analogy a bit. One interesting consequence of the existence of level 2 is that for some finite groups, the most natural (read: easiest to construct) representations are infinite dimensional, and one can reasonably argue using lattice vertex algebras that this holds for some exceptional families as well. John Duncan has some recent work constructing structured vertex superalgebras whose automorphism groups are sporadic simple groups outside the happy family.

I think one interesting question that has not been suggested by other responses (and may be too open-ended for MO) is why the monster has no small representations. There are no faithful permutation representations of degree less than $9 \times 10^{ 19 }$ and there are no faithful linear representations of dimension less than 196882. Compare this with the cases of the numerically larger groups $A_{50}$ and $E_8(\mathbb{F}_2)$, where we have linear representations of dimension 49 and 248. This is a different sense of hugeness than in the original question, but one that strongly impacts the computational feasibility of attacking many questions.

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The sporadic finite simple groups aren't that big! The Monster group is smaller than the alternating group on 50 letters.

I think a better place for you to start (rather than wondering about general intuition for very particular objects) would be to look at Griess's Twelve Sporadic Groups. My recollection is that that book is quite accessible, and it'll let you get a glimpse of what's going on here (mostly through the smallest examples of sporadic groups rather than the largest).

Another good place to start would be to understand the exceptional Lie groups. They're similar in spirit, but much easier to understand.

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Just to comment on sizes of groups: the symmetric group on 52 letters arises "naturally" in the study of shuffling the standard deck of cards. –  Michael Lugo Mar 9 '10 at 17:06
    
I guess I should have been more specific. So far my intuition tells me that groups that are simple, and maybe this is suggested by the nomenclature, should be smaller than groups which are not simple, simply because a group which is not simple appears to have richer structure. From this perspective, the existence of a finite simple group like Monster, with combinatorially large order and a superficially random integer factorization of that order, is baffling. I will check out that book though. Hopefully my library has it. –  REDace0 Mar 9 '10 at 18:48
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Non-simple groups can have much less structure than simple groups of comparable order. Consider, for example, large cyclic groups, or more generally groups that are the direct product of their Sylow subgroups. And simple groups can have a lot of structure - normal subgroups are not the only thing that constitutes "structure"! –  Qiaochu Yuan Mar 9 '10 at 19:07
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In another way, sporadic finite simple groups are extremely large, compared to our understanding of general groups. A sylow subgroup of size $2^{46}$? No one even knows how many p-groups of size $p^8$ there are, let alone what they are, and the Higman PORC Conjecture is still a conjecture. –  Ben Mar 9 '10 at 20:22
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@Ben The Sylow 2-subgroups of the finite simple groups are very highly studied and quite well understood. Brauer's approach, which was very successful, started by considering an involution and its centralizer. Many of the classification results were something like, "Any finite simple group whose Sylow 2-subgroup is abelian/dihedral/A wreath B must be of the following form: X(q), or of order n." –  Douglas Zare Mar 10 '10 at 11:39

I am not an expert, but... First, not all of the sporadic groups are that huge -- the smallest Mathieu group, for instance, has order about 8000 (this is off the top of my head), and I think the Janko groups have order on the order of $10^6$ or $10^7$.

Second, a lot of the sporadic groups are connected with the automorphism group of the Leech lattice -- and one would expect a "very symmetric" 24-dimensional lattice to have a really big automorphism group! So to some extent I think there's just a combinatorial explosion because 24 is small but 24!, for instance, is pretty big. Also keep in mind that, to some people (Ramsey theorists, analytic number theorists), $10^{53}$ is tiny. It's all relative.

Finally, as Sonia pointed out, you can ask "why are there sporadic simple groups at all?" The most satisfying answer would seem to me to be that the way we think about the classification of finite simple groups is wrong, in which case the sizes of the sporadic groups might just be an accident of history.

P.S. This would probably be better in a comment, but don't worry about being "just" a graduate student -- I'm an undergraduate myself, and I'm far from the only one. We've even had some contributions from high schoolers on MO. If your questions and/or answers are good, it doesn't matter if you're a cockroach.

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I disagree with the idea that the way we classify finite simple groups is the reason there are sporadic finite simple groups. There infinite families are cyclic groups of prime order, alternating groups, and groups of Lie type over a finite field. (In fact, there should be some way to view the alternating groups as degenerate groups of Lie type over a field with one element.) The sporadic simple groups don't fit into these categories. If we meet alien mathematicians, they will agree that these are sporadic. –  Douglas Zare Mar 9 '10 at 16:57
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I wouldn't be quite so confident. Take the example of Lie algebras. If someone manages to give a good construction of the Vogel plane (and the universal Lie algebra object) then the exceptional Lie algebras start to look a lot less exceptional. What if aliens knew about finite type knot invariants before Lie algebras? –  Noah Snyder Mar 9 '10 at 17:07
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I really want to meet an alien mathematician so I can see if they actually agree with us about the things we say they'll agree with us about. (I agree with you here, but I'm not an alien.) –  Michael Lugo Mar 9 '10 at 17:08
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@Douglas: I'd like to play the alien devil's advocate. Suppose I am totally uninterested in vector spaces over finite fields but am very interested in some other set of structures with interesting automorphism groups (say block designs, to fix ideas) and I classify the "classical" simple groups this way. Is there any reason this classification should agree with the classification of groups of Lie type? –  Qiaochu Yuan Mar 9 '10 at 17:16
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@Qiaochu Yuan: There are deep reasons for collecting together the finite groups of Lie type, not explained by the elementary viewpoint that they are described as "matrix groups over finite fields". By thinking in structural terms (e.g., $BN$-pairs), one can dispose of many properties of these groups (e.g., prove their simplicity!) by a single argument. It is as if there is only one connected reductive group $G$, and often one can avoid explicit calculations until reducing to some special case with ${\rm{SL}}_2$ or ${\rm{PGL}}_2$. The techniques don't easily apply to sporadics. –  BCnrd Mar 9 '10 at 20:22

The primes dividing the order of the Monster group are precisely the primes $p$ so that the surface $\mathbb H^2/\Gamma_0(p)^*$ has genus $0$, as was observed by Ogg. See Monstrous Moonshine, which was about the deep connections between number theory and the Monster group. I hope an expert elaborates.

According to Peter McMullen, regular polytopes are "wayside shrines at which one should worship on the way to higher things." Further along the road are E8, the Leech lattice, and then the Monster group.

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Indeed the question is too vague for a precise answer, but nevertheless somehat natural ;-)

I want to give some more details and clearifications to the "hierarchy", that has been broached by Carnahan above. The "generic" simple groups are the Lie type groups of arbitrary size and the alternating groups. Furthermore, the main induction step of the classification theorem was, that the centralizer of an involution of a simple group (an order-2-element/involution is the "only" thing we have "a-priori" in an arbitrary simple group by Feit-Thomson) is close (!) to simple. So there is the chance of sporadic group branching off from a Lie-Type or alternating group and inductively proceed for some steps until it terminates.

This inductive process of constructing a much larger simple group from it's involution centralizer being a prescribed (already large) simple group in extremely rare (!) situations could be thought of some sort of answer to your question. It is by the way one reason for the incredible length of the classification result (a tremendous case-by-case argument)...and for my personal view on the meta-debate above, that sporadics are more sporadic (not more unnatural!) than others, as much to us as to species 8472 ;-) ;-)

Most examples go only one step (and still are very large!), e.g. almost all so-called pariahs:

  • $J_1,J_3\leftarrow A_5$
  • $Ly \leftarrow A_{11}$
  • $ON \leftarrow SL_3(4)$
  • $Ru \leftarrow\;^2SO_5(8)$ "twisted" Lie-type (alike the unitaries over finite fields)
  • $J_4\leftarrow M_{22}\leftarrow\ldots$ branches off already one induction step beyond Lie (see below)

Note that most of these cases already appear as involution centralizers of Lie-type groups, which is somewhat miraculous and was often the reason to study this particular class and find only distinct "irradic" different choices. E.g. $^2G(3^{2n+1})\leftarrow SL_2(3^n)$ and the only other possible case $SL_2(4)\cong SL_2(5)\cong A_5$ lead Janko 1965 to the first new sporadic $J_1$ in almost a century.

On the other hand there is a VERY remarkable string of induction steps to the Monster and with modifications to the other sporadic groups "involved" in it. It goes roughly as

$M\leftarrow Co_1 \leftarrow M_{24}\leftarrow SL_3(4)$

and heavily relies on the already mentioned Golay-Code resp. Steiner System $S(24,8,5)$ - beautiful, very sporadic and purely combinatoric objects! Along the induction steps, the combinatorical objects with these groups as automorphism groups can be extended as well, very roughly like

Griess-Algebra $\leftarrow$ Leech-Lattice $\leftarrow$ Steiner-System $\leftarrow$ Projective-Plane

One striking numerical reason for this construction and the very exotic behaviour to work exactly for $24$ dimensions (also responsible for the $2^{24}$-factor mentioned above) is:

$1^2+2^2+\ldots+23^2+24^2=70^2$

This is provably impossible for larger numbers (by hard number theory) and is the striking numerical coincidence used in the side-by-side construction of the Golay code, the Steiner System and the $24$-dimensional Leech lattice, which is the most dense sphere packing of all dimensions and the reason e.g. the kissing number is known for this dimension!

Hope that gives some intuition and "personality" for the various sporadics ;-) ;-)

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