Timeline for How many sporadic simple groups are there, really?
Current License: CC BY-SA 3.0
15 events
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| S Aug 21, 2017 at 11:01 | history | suggested | user21230 |
Added tag sporadic-groups
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| Aug 21, 2017 at 10:39 | review | Suggested edits | |||
| S Aug 21, 2017 at 11:01 | |||||
| Dec 11, 2013 at 14:18 | comment | added | Simon Rose | Fair enough. I was mostly asking (and perhaps I didn't word it well) what properties the group had that justified its inclusion in the infinite family. From some of the comments though, it sounds like the line between sporadic and non-sporadic may be a little blurry with some of the smaller groups. | |
| Dec 11, 2013 at 14:12 | comment | added | Nick Gill | I can't see how you're going to get a definitive answer to this question - it all depends on what you mean by 'sporadic'. Perhaps it would be as well to say that certain groups 'exhibit sporadic behaviour', rather than 'are sporadic'. For example, as Noam says, $A_5$ has several other incarnations both of which are associated with a BN-pair. So $A_5$ exhibits sporadic behaviour because it is alternating AND has two different BN-pair structures! Is that more or less sporadic than ${^2F_4}(2)'$ which is of Lie type but has no BN-pair? Depends on who's asking I guess... | |
| Dec 11, 2013 at 5:27 | comment | added | Noam D. Elkies | Also $^2 G_2(3)$. | |
| Dec 11, 2013 at 5:26 | comment | added | Pete L. Clark | Also, a look at the tables of finite group of Lie type on wikipedia shows that there is often some funny business at $2$: sometimes the groups are too small to be simple; more pertinently, the phenomenon that you have to pass to the derived subgroup (which has index $2$) to get a simple group occurs also for $B_2(2)$ and $G_2(2)$. (Added: I stopped reading before I got to the end of the list! Noam points out a funny thing that happens at $3$.) | |
| Dec 11, 2013 at 5:25 | comment | added | Noam D. Elkies | ... The Tits group is a wash, because it compensates for the non-simple ${}^2F_4({\bf Z}/2{\bf Z})$. It would be nice if the total discrepancy balanced out to zero. (Does it?) | |
| Dec 11, 2013 at 5:21 | comment | added | Noam D. Elkies | Another way to do the count ends up with much less than $26$ or $27$: each sporadic simple group not in one of the infinite families adds one, but each group in any infinite family that isn't simple subtracts one, as does each small coincidence between groups in two different families. For instance $A_n$ ($n \leq 4$) contributes $-4$, $\text{PSL}_2({\bf Z}/2{\bf Z})$ and $\text{PSL}_2({\bf Z}/3{\bf Z})$ contribute $-1$ each, and $\text{PSL}_2({\bf Z}/5{\bf Z}) \cong \text{PSL}_2({\bf F}_4) \cong A_5$ contributes $-2$ more. [cont'd] | |
| Dec 11, 2013 at 5:17 | comment | added | Pete L. Clark | Anyway, the point of my remark is that (so far as I see it, obviously) the classification theorem for finite simple groups is not really "There are precisely $18$ infinite families and $26$ sporadic simple groups." It is "Although there are infinitely many finite simple groups, we can describe them in a very explicit, finite way: namely..." Even the five families have some overlap...but that's just a foible of our description of the classification, it seems to me. | |
| Dec 11, 2013 at 5:14 | comment | added | Pete L. Clark | @Qiaochu: Sure, and the value of the cyclotomic character at (any) complex conjugation is $-1$. (Surely these facts are related...) I didn't say anything for or against Conway's nomenclature but rather that it doesn't matter whether we use it or not. Maybe you're suggesting that it helps people think about Frobenius elements at the (ahem) infinite prime, but that seems a little much to me. | |
| Dec 11, 2013 at 4:20 | comment | added | Qiaochu Yuan | @Pete: there's at least one reason I can think of to like the name $-1$ for the infinite prime, which is that it suggests the correct way to compute the (compactly supported) Euler characteristic of the real points of certain nice varieties $X$ defined over $\mathbb{Z}$. Namely, if $|X(\mathbb{F}_q)|$ is a polynomial in $q$ (and maybe some other hypotheses), then substituting $q = -1$ gives this Euler characteristic. | |
| Dec 11, 2013 at 0:51 | comment | added | Pete L. Clark | I don't mean this in a confrontational way at all, but: does it really matter whether we view this group as sporadic or not? If so, why? (By way of comparison, Conway likes to talk about "the prime $-1$" rather than the infinite prime. I am confident that it doesn't matter whether we say it that way or not.) | |
| Dec 10, 2013 at 22:23 | comment | added | Gerry Myerson | I remember noticing at some point in the 1970s that the number of known sporadic simple groups was equal to the number of known Mersenne primes. Since then, the group theorists have fallen way behind the number theorists. | |
| Dec 10, 2013 at 18:20 | comment | added | Derek Holt | It's included in that family because it's equal to $^2F_4(2)'$. So it's not surprising that people don't agree about whether it's sporadic or not. | |
| Dec 10, 2013 at 18:02 | history | asked | Simon Rose | CC BY-SA 3.0 |