### 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 Society7 (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 $1024 = 2^{10}$. A finite group is called a
‘$2$-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

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

as $N \to \infty$?

Posted at November 28, 2012 8:47 PM UTC
## Re: Almost All of the First 50 Billion Groups Have Order 1024

‘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.