Almost All of the First 50 Billion Groups Have Order 1024

Posted by Tom Leinster

Here’s an incredible fact: of the 50 billion or so groups of order at most 2000, more than 99% have order 1024. This was announced here:

Hans Ulrich Besche, Bettina Eick, E.A. O’Brien, The groups of order at most 2000. Electronic Research Announcements of the American Mathematical Society 7 (2001), 1–4.

By no coincidence, the paper was submitted in the year 2000. The real advance was not that they had got up to order 2000, but that they had ‘developed practical algorithms to construct or enumerate the groups of a given order’.

I learned this amazing nugget from a recent MathOverflow answer of Ben Fairbairn.

You probably recognized that 1024 = 2 10 1024 = 2^{10} . A finite group is called a ‘ 2 2 -group’ if the order of every element is a power of 2, or equivalently if the order of the group is a power of 2. So as Ben points out, what this computation suggests is that almost every finite group is a 2-group.

Does anyone know whether there are general results making this precise? Specifically, is it true that

number of 2-groups of order ≤ N number of groups of order ≤ N → 1 \frac{\text{number of 2-groups of order } \leq N}{\text{number of groups of order } \leq N} \to 1

as N → ∞ N \to \infty ?

Posted at November 28, 2012 8:47 PM UTC