30
$\begingroup$

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple groups have a smallish number of conjugacy classes relative to size. Even with this intuition though, some of the numbers appearing in the list of sporadic simple groups are a bit surprising. For instance, the Monster group, with more than $10^{53}$ elements and $15$ prime divisors, has fewer than $200$ conjugacy classes. (An alternating group of comparable order has more than $30 000$ conjugacy classes.) $M_{22}$ has $443520$ elements and only $12$ conjugacy classes.

What is going on here? Is there something about the special combinatorial structures that allow these groups to exist that also makes the centralisers exceptionally small?

$\endgroup$
  • 9
    $\begingroup$ An equivalent question is, why do sporadic simple groups lack small representations? $\endgroup$ – S. Carnahan Dec 14 '14 at 23:56
  • 10
    $\begingroup$ @S.Carnahan : Well, it's maybe not quite equivalent. Having few conjugacy classes is equivalent to the mean irreducible character degree being large. For example, $M_{12}$ has an irreducible complex representation of degree $11$, Suzuki's sporadic group of order 448,345,497,600 has a $12$-dimensional irreducible complex representation. $\endgroup$ – Geoff Robinson Dec 15 '14 at 0:15
  • 6
    $\begingroup$ The fact that the Monster needs nearly 200000 dimensions to be represented linearly over any field is striking in its own right, but only a few of the sporadics are like this. $\endgroup$ – Colin Reid Dec 15 '14 at 0:20
  • 2
    $\begingroup$ For which sporadic simple groups have you actually checked that there is no group (whether simple or not) of the same or larger order with fewer conjugacy classes? $\endgroup$ – Stefan Kohl Dec 15 '14 at 15:00
17
$\begingroup$

This is also rather an expanded comment. -- Since for purely arithmetical reasons, $\ln(\ln(|G|))$ is a lower bound for the number $k(G)$ of conjugacy classes of a finite group $G$, maybe $$ f(G) := \ln(k(G))/\ln(\ln(\ln(|G|))) $$ is a better measure than $k(G)/|G|$ for how many or how few conjugacy classes a group $G$ has in comparison with its order. For example we have (examples ordered by group order):

  • $f({\rm A}_5) \approx 4.68799$,

  • $f({\rm PSL}(2,7)) \approx 3.64930$,

  • $f({\rm A}_6) \approx 3.39930$,

  • $f({\rm PSL}(2,8)) \approx 3.64187$,

  • $f({\rm PSL}(2,11)) \approx 3.3204$,

  • $f({\rm A}_7) \approx 3.04392$,

  • $f({\rm PSL}(3,3)) \approx 3.23520$,

  • $f({\rm M}_{11}) \approx 2.92936$,

  • $f({\rm PSL}(2,31)) \approx 3.53994$,

  • $f({\rm A}_8) \approx 3.17897$,

  • $f({\rm PSL}(3,4)) \approx 2.77366$,

  • $f({\rm Sz}(8)) \approx 2.83466$,

  • $f({\rm M}_{12}) \approx 3.03727$,

  • $f({\rm J}_1) \approx 2.96687$,

  • $f({\rm A}_9) \approx 3.16283$,

  • $f({\rm M}_{22}) \approx 2.63787$,

  • $f({\rm J}_2) \approx 3.20085$,

  • $f({\rm A}_{10}) \approx 3.23851$,

  • $f({\rm M}_{23}) \approx 2.76986$,

  • $f({\rm A}_{11}) \approx 3.31013$,

  • $f({\rm Sz}(32)) \approx 3.39405$,

  • $f({\rm HS}) \approx 3.01600$,

  • $f({\rm J}_3) \approx 2.88256$,

  • $f({\rm M}_{24}) \approx 3.00146$,

  • $f({\rm PSL}(6,2)) \approx 3.55208$,

  • $f({\rm O'N}) \approx 2.85566$,

  • $f({\rm Fi}_{22}) \approx 3.36345$,

  • $f({\rm HN}) \approx 3.18141$,

  • $f({\rm B}) \approx 3.54764$,

  • $f({\rm A}_{43}) \approx 6.61233$,

  • $f({\rm M}) \approx 3.34883$,

  • $f({\rm A}_{44}) \approx 6.69491$.

$\endgroup$
  • 2
    $\begingroup$ This list is very interesting. It suggests to me almost the opposite of what the OP asked, namely that the number of conjugacy classes in a simple group is a very well-behaved statistic. I wonder whether one could prove an absolute upper bound for your function $f(G)$, as $G$ ranges over all of the (non-alternating) simple groups? This seems too much to ask, but you never know... I presume that $f(G)$ is unbounded for $A_n$? Could one also prove that $f(G)$ takes its minimal value when $G=M_{22}$? $\endgroup$ – Nick Gill Dec 15 '14 at 20:23
  • $\begingroup$ @NickGill: We have indeed $\lim_{n \rightarrow \infty} f({\rm A}_n) = \infty$, as one can check using the well-known approximation formulas for $n!$ and the partition function $p(n)$. As to your other questions, I don't know. $\endgroup$ – Stefan Kohl Dec 15 '14 at 21:10
  • $\begingroup$ @NickGill: At least we also have $\lim_{p \rightarrow \infty} f({\rm PSL}(2,p)) = \infty$. $\endgroup$ – Stefan Kohl Dec 15 '14 at 21:46
  • $\begingroup$ Yes, that makes sense as an invariant. I suppose it is only some of the sporadics that stand out. Maybe the issue is more that $A_n$ has unusually many classes by simple group standards? $\endgroup$ – Colin Reid Dec 15 '14 at 22:01
  • 1
    $\begingroup$ @ColinReid: The former yes, but the latter I doubt: I computed the approximate values $f({\rm PSL}(n,2))$ for $n = 3, 4, \dots, 32$. -- They are $3.6493, 3.17897, 3.22354, 3.55208, 3.81153, 4.1534, 4.46381, 4.79852, 5.11825, 5.45075, 5.77278, 6.10037, 6.42311, 6.74737, 7.06869, 7.39041, 7.71006, 8.02942, 8.34732, 8.66458, 8.98068, 9.29605, 9.61042, 9.92402, 10.2367, 10.5487, 10.8599, 11.1703, 11.48, 11.789$. For larger $n$, increasing $n$ by $1$ seems to add about $0.30$ to $0.33$. $\endgroup$ – Stefan Kohl Dec 15 '14 at 23:31
12
$\begingroup$

This isn't really an answer, but an extended comment that might shed a little light.

Let $k(G)$ be the number of conjugacy classes in a group $G$, and let me interpret your question as a query about the proportion $k(G)/|G|$. The following paper suggests that the behaviour you describe for sporadic groups is actually shared by the finite groups of Lie type too.... but that the proportion $k(G)/|G|$ "takes longer" to reach zero.

Liebeck, Martin W.; Pyber, László Upper bounds for the number of conjugacy classes of a finite group. J. Algebra 198 (1997), no. 2, 538–562.

The main result of this paper is that $k(G)\leq (6q)^{\ell}$ for a group of Lie type of untwisted rank $\ell$ over a field of $q$ elements.

In the case, for instance, where $G={\rm PSL}_{\ell+1}(q)$, we know that $|G|>q^{\frac12\ell^2}$ for $q$ big enough, so one obtains immediately that (for $\ell>2$) $\lim\limits_{|G|\to\infty} \frac{k(G)}{|G|}=0$. When $\ell=1, 2$, one can check directly that the same is true.

Indeed, for the other families, one has similar lower bounds on the size of $|G|$, except that one might need to replace the $\frac12$ by some other constant. Thus, in general, it is true that for $G$ in the family of simple groups of Lie type $$\lim\limits_{|G|\to\infty} \frac{k(G)}{|G|}=0.$$ Thus the groups of Lie type are exhibiting exactly the same behaviour asymptotically as what you observe for sporadic groups.

Of course, I have chosen just one interpretation of your question, and even from this perspective, this is only a partial answer: I cannot explain why the sporadic groups reach such small values for $\frac{k(G)}{|G|}$ so much more quickly than the groups of Lie type. In truth, though, I doubt an "answer" such as this really exists, since saying anything about the general behaviour of sporadic groups seems incredibly difficult.

Added later: Actually, I just looked at the ATLAS and I must say I don't see a great deal of evidence that the sporadic simple groups in general have much fewer conjugacy classes than groups of Lie type of similar sizes. For instance:

  • $k(M_{24})=26, \, \, k(G_2(4))=32$
  • $k(HN)=54, \, \, k(Fi_{22})=65, \, \, k(F_4(2))=95$
  • $k(O'N)=39, \, \, k(C0_3)=42, \, \, k(O_{10}^+(2))=97$

In each row, groups have comparable size. In each case the sporadic group(s) have a few less conjugacy classes but not by a great deal. Perhaps your question should really be about the Monster?

$\endgroup$
  • 6
    $\begingroup$ As a matter of interest, It is proved in a 2006-ish paper of Bob Guralnick and myself that in general, $\frac{k(G)}{|G|} \to 0$ as $[G:F(G)] \to \infty$ ( for finite $G$). $\endgroup$ – Geoff Robinson Dec 15 '14 at 17:07
  • $\begingroup$ @GeoffRobinson, this is interesting. Indeed, given that this phenomenon is so general, the rate of convergence becomes interesting when considering the "non-abelian-ness" mentioned in the original question. The Liebeck-Pyber paper is very strong in this regard, but of course such a notion makes no sense in connection with the sporadic groups. $\endgroup$ – Nick Gill Dec 15 '14 at 17:12

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.