The n-Category Café

‘Some people’ think the right way to count objects in a category is with mass formulas, where each object $G$ has a weighting of $1/|\mathrm{Aut}(G)|$. I wonder how much this changes your stats.

I’d like to take this moment to advertise the (old) book “Groups of order 2^n, n \leq 6”.

This isn’t surprising some how to me. 99% of all semigroups have a zero element and satisfy xyz=0. Conjecturally most of these have trivial automorphism groups and so even Jim’s solution of weighting doesn’t help. I think p-groups are the analog in group theory of nilpotent semigroups (in the above sense). It is generally believed that classification of p-groups up to isomorphism is essentially impossible. I would suspect most p-groups have small isomorphism groups, but I don’t know for sure.

Is this just a property of small prime numbers? Of the small number of groups that are not 2-groups, do most of them have orders divisible by 6? (The link doesn’t resolve for me right now, so I can’t look at the announcement itself.) Or is there something special about “the oddest prime” other than that it’s the smallest one?

Ronald Solomon reported this back in 2001 here:

An ever-improving arsenal of computer algorithms for the identication of permutation groups, linear groups and more generally “black box” groups is being assembled by Kantor, Seress and many other researchers. In particular there is hope of determining all finite groups of order at most 2001 in the year 2001. On the other hand, calculation of groups of order $2^10$ has reconfirmed the long-known fact that “most” finite groups are nilpotent groups of nilpotence class at most 2 and they exist in frightening numbers.

I like what he goes on to say next, and quoted it partly in my book,

Thus the classication of all finite groups is completely infeasible. Nevertheless experience shows that most of the finite groups which occur “in nature”– in the broad sense not simply of chemistry and physics, but of number theory, topology, combinatorics, etc.– are “close” either to simple groups or to groups such as dihedral groups, Heisenberg groups, etc., which arise naturally in the study of simple groups. And so both the methodologies and the database of information generated in the course of the Classification Project remain of vital importance to the resolution of questions arising in other disciplines. (p. 347)

I’m also unsurprised by the result. Anyone who’s spent some time in an algebra class classifying groups of small orders has noticed that the complexity increases a lot when the multiplicities of prime factors rises. Notably $log_2 (2000) > 10$, but $log_3 (2000) < 7$ and $log_5 (2000) < 5$.

Aside from the question of how it’s appropriate to weight groups, this raises the question about how one should order the orders. That is, does it make more sense to ask a question about all groups with order at most $N$, or to ask about (say) all groups with at most $M$ prime factors (with multiplicity) and each prime factor less than or equal to $P$?

More simply, it would be interesting to compare the number of groups of order $p^k$ for fixed $k$ and increasing prime $p$. Does anyone happen to know the numbers of groups of order $3^n$, $n \le 6$?

Cool! Turns out the number of groups of order $p^n$ for prime $p$ is

$$p^{2n^3/27 + O(n^{8/3})}$$

And, you can see a nice table of how many groups there are of orders $\le 2000$ here.

I wonder if anyone’s done these counts for small finite categories.