Experimenting with finite groups recently, I’ve learnt a few things I was completely unaware of:

(1) there are so many distinct groups of order 512 (or 1024, 1152, 1536 and 1920) up to isomorphism, that Magma and GAP are not able to recognize them (they have databases of groups of order up to 2000, except for 1024, but given one abstract group of this order, they can not pinpoint which element of the database it is). Precisely, there are 10494213 distinct finite groups of order 512 up to isomorphism, 157877 of order 1152, 408641062 of order 1536 and 241004 of order 1920. These numbers are several order of magnitude larger than what I would have guessed if asked point-blank.

(2) the order of groups of a fixed Lie type, say *X(q)*, where *X* could be *PSL*, is not always monotonic with respect to the variable *q*; for instance:

gap. Order(PSL(3,17));

6950204928

gap. Order(PSL(3,19));

5644682640

gap. Order(PSL(3,29));

499631102880

gap. Order(PSL(3,31));

283991644800

(Of course, this is then easy to check, and the point is that when *q-1* is divisible by 3, the order drops as the center of *SL(3,q)* becomes a bit bigger).

Is it true that limit_{n \to \infty} #{groups of order 2^k <= n}/#{groups of order <= n} = 1?

It’s possible, but I don’t know. I haven’t looked in the literature, and although I know there’s a fair amount which is known about the number of abelian groups of a given order, I have no idea about what is known of the general groups.

Indeed so it seems. See under the heading “Prevalence” in the wikipedia entry on “p-group”. The reference for this fact given therein is

Besche, Hans Ulrich; Eick, Bettina & O’Brien, E. A. (2002), “A millennium project: constructing small groups”, International Journal of Algebra and Computation 12(5): 623–644, MR1935567, ISSN 0218-1967

Thanks for the reference! The paper is available at

http://www-public.tu-bs.de:8080/~hubesche/pl.html

and seems quite informative about this type of things, although I can’t find there (quickly) a precise statement of the type suggested, just an upper bound due to Pyber on the number of isomorphism classes of groups of order n.

Having read a bit more carefully the paper, it really seems that it does not refer precisely to the property suggested by Comment 1.

It seems to me, after searching the web for a bit, that the speculation that almost all (isomorphism classes of) finite groups are 2-groups (in the sense of comment 1) is as yet unproven. A paper by L. Pyber titled “Enumerating finite groups of given order” published in Annals of Math. in 1993 makes the conjecture that almost all finite groups are nilpotent, which seems to be open still.

I shall dispute the wikipedia entry mentioned above.

Sims showed in 1965 that there are p^(2n^3/27 + O(n^(8/3))) isomorphism types of groups of order p^n, and the primary contributors are three-step nilpotent. The exponent has the same asymptotics as the dimension of the moduli space of commutative algebras of finite rank, and Poonen has a paper about this on his web page. I saw him give a talk about this last year, and he said that the problem in comment #1 was still open.

A corollary of the results of Higman Sims and myself is that the number of groups of order at most n is “exactly”

n^((2/27+o(1))(log n)^2)).

Hence the number of groups of order at most n is roughly the same as the number of 2-groups of order at most n.

Actually in my old paper I show that the number of groups of order n with given Sylow subgroups is much smaller.

Using this one can easily prove that

most groups of order at most n have a normal 2 subgroup of index at most f(n) where f(n)=o(n).To prove that,say f(n)=0(1) seems to require a much better understanding of the case of p-groups.

By the way there is a recent book by Blackburn-Neumann-Venkataraman on “Enumeration of finite groups”

Cambridge Tracts in Mathematics,173

Thanks very much for this information! (and for proving such a fascinating result…)

Oops, the result on random groups I mentioned above (while true) is not too interesting. A better version is that most groups of order at most n have a normal 2 subgroup of index at most f(n) where f(n)=n^o(1).