Timeline for How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jun 14, 2010 at 5:33 | comment | added | S. Carnahan♦ | @David Lehavi: It seems unlikely that you'll read this comment now, but I might as well jot it down. The weighted enumeration specifies that (Z/2)^3 has weight 1/168, and the dihedral group has weight 1/8. I was pointing out that the sum of these weights was at most 1 for small orders. | |
| Apr 15, 2010 at 6:06 | comment | added | David Lehavi | @Scott, Brian: even in order 8 you have two weighted groups: (Z/2)^3 has GL_3(2) outer ones, whereas D_4 has an outer involution, and a Klien group of inner involutions. | |
| Apr 14, 2010 at 6:41 | comment | added | S. Carnahan♦ | @Theo: I think that game would have more mathematical content if the players had to describe each of the groups, rather than reciting lists of numbers. (I'm not claiming that the revised game is particularly interesting, though.) | |
| Apr 14, 2010 at 5:36 | comment | added | Theo Johnson-Freyd | There is a favorite game, played by two players. I'll go first. On my turn, I say how many groups there are of size 1: "there is one group of size 1." Now you say how many groups are of size 2: "there is one group of size 2." Me: "there is one group of size 3." You: "there are two groups of size 4." And so on. Whoever doesn't know an answer first loses. The trick is that most people lose at size 16, and even if they don't, you'll certainly lose at 32 if not before. | |
| Apr 14, 2010 at 5:24 | answer | added | Gerhard Paseman | timeline score: 5 | |
| Apr 14, 2010 at 1:50 | answer | added | Richard Stanley | timeline score: 6 | |
| Apr 14, 2010 at 1:37 | answer | added | JS Milne | timeline score: 18 | |
| Apr 14, 2010 at 1:14 | comment | added | Joel David Hamkins | Brian, that is an interesting question, but for the application I mentioned to studying the hierarchy of classification problems, the question I ask here seems to be the relevant one, even if the functions are bumpy. | |
| Apr 14, 2010 at 0:55 | answer | added | Kevin O'Bryant | timeline score: 7 | |
| Apr 14, 2010 at 0:40 | comment | added | S. Carnahan♦ | Brian, that is a very interesting question (and it reminds me of the nice enumeration of ss elliptic curves). There aren't any orders less than 16 with more than one weighted group, and I'm unwilling to work out more by hand. | |
| Apr 14, 2010 at 0:31 | answer | added | S. Carnahan♦ | timeline score: 10 | |
| Apr 14, 2010 at 0:29 | comment | added | BCnrd | Wouldn't it make a lot more sense to weight things by the size of their automorphism group? (I know, it's not the same question, but is quite natural, and may have a chance at being less "bumpy".) I wonder if anyone has attacked it. Certainly for the count of finite extensions of a $p$-adic field, the answer is much more elegant with such weighting factors. | |
| Apr 14, 2010 at 0:10 | comment | added | Kevin Buzzard | "Most finite groups are 2-groups", so G(n) will be a very bumpy function. I almost want to say that it's not advisable to try and approximate it by some "simple" function like x^alpha or e^(x^alpha). To give an example: there are 11759892 groups of order at most 1023, and 49487365422 of order 1024: this is about 4000 times bigger! The number of groups of size p^e is about p^((2/27)e^3 and presumably the p=2 terms dominate. This should be enough information to see how G(n) is growing. | |
| Apr 13, 2010 at 23:47 | answer | added | François G. Dorais | timeline score: 8 | |
| Apr 13, 2010 at 23:37 | answer | added | Michael Lugo | timeline score: 9 | |
| Apr 13, 2010 at 23:22 | answer | added | François G. Dorais | timeline score: 14 | |
| Apr 13, 2010 at 23:14 | answer | added | Rob Harron | timeline score: 17 | |
| Apr 13, 2010 at 22:43 | comment | added | Qiaochu Yuan | For graphs, see research.att.com/~njas/sequences/A000088 . | |
| Apr 13, 2010 at 22:08 | comment | added | Joel David Hamkins | From OEIS: Number of groups of order n: research.att.com/~njas/sequences/A000001 | |
| Apr 13, 2010 at 22:01 | comment | added | Qiaochu Yuan | Have you checked to see whether asymptotics are available on the OEIS? | |
| Apr 13, 2010 at 22:00 | comment | added | Hailong Dao | Great question(s)! | |
| Apr 13, 2010 at 21:57 | history | asked | Joel David Hamkins | CC BY-SA 2.5 |