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In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.

Is this for real? How can one deduce this result? Is there a nice way or do we just check all finite groups up to isomorphism?

Thanks!

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    $\begingroup$ There are $49487365422$ of order 1024 $\endgroup$
    – PAD
    Nov 20, 2012 at 15:32
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    $\begingroup$ No doubt this about isomorphism classes of groups. $\endgroup$
    – Marc van Leeuwen
    Nov 20, 2012 at 17:28
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    $\begingroup$ @yatima2975: From this article : "gnu(2048) is still not precisely known, but it strictly exceeds 1774274116992170, which is the exact number of groups of order 2048 that have exponent-2 class 2, and can confidently be expected to agree with that number in its first 3 digits." $\endgroup$
    – Mikko Korhonen
    Nov 21, 2012 at 10:59
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    $\begingroup$ Is there a chart (or database to build one) to visualize the distribution of group orders less than $n$ for individual $n$ up to ~2000? $\endgroup$
    – alancalvitti
    Nov 22, 2012 at 17:20
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    $\begingroup$ @alancalvitti: Try this one oeis.org/A000001/b000001.txt $\endgroup$
    – Mikko Korhonen
    Nov 23, 2012 at 7:50

3 Answers 3

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Here is a list of the number of groups of order $n$ for $n=1,\ldots,2015$. If you add up the number of groups of order other than $1024$, you get $423{,}164{,}062$. There are $49{,}487{,}365{,}422$ groups of order $1024$, so you can see the assertion is true. (In fact the percentage is about $99.15\%$.)

As far as I know there is no reasonable way to deduce a priori the number of isomorphism classes of groups of a given order, though I believe that combinatorial group theory has some methods for specific cases. A general rule of thumb is that there are a ton of $2$-groups, and in fact I have heard it said that "almost all finite groups are $2$-groups" (though I cannot cite a reference for this statement).

EDIT: As pointed out in the comments, "almost all finite groups are $2$-groups" is still a conjecture. There is an asymptotic bound on the number of $p$-groups of order $p^n$, however. Denoting by $\mu(p,n)$ the number of groups of order $p^n$, $$\mu(p,n)=p^{\left(\frac{2}{27}+O(n^{-1/3})\right)n^3},$$ which is proven here. This colossal growth along with the results of Besche, Eick & O'Brien seem to be what primarily motivated the conjecture.

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  • $\begingroup$ A while ago I tried to find a reference for this "almost all..." result. I think it is just a folklore statement, with the paper which is the subject of this thread proffered as evidence. $\endgroup$
    – user1729
    Nov 20, 2012 at 15:42
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    $\begingroup$ (I wonder if there are more groups of order $3^{10}$ than of order $2^{10}$? Genericity proofs are...unsavoury...at least to my pallet...) $\endgroup$
    – user1729
    Nov 20, 2012 at 15:44
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    $\begingroup$ Of course it's possible to deduce the number of isomorphism classes of groups of (finite) order $n$: write down all possible $n$ by $n$ multiplication tables, check which satisfy the group axioms, check every bijection between each pair to see if it's a group isomorphism. Since everything is finite, this can all be computed in finite time. The hard part is finding ways to do it in a sane amount of time. $\endgroup$
    – Chris Eagle
    Nov 20, 2012 at 15:55
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    $\begingroup$ According to the list linked in the answer, there are 504 groups of order $3^6=729$ and 267 groups of order $2^6=64$. There are 15 groups of order $5^4=625$ and also of order $3^4=81$ and 14 groups of order $2^4=16$. $\endgroup$
    – Mark Bennet
    Nov 20, 2012 at 16:40
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    $\begingroup$ Almost all groups are infinite. $\endgroup$
    – Marc van Leeuwen
    Nov 20, 2012 at 17:29
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This is true. The amount of groups of order at most 2000 (up to isomorphism) was calculated precisely for the first time in 2001 by Besche, Eick and O'Brien. Here is the announcement of their result:

We announce the construction up to isomorphism of the $49 910 529 484$ groups of order at most $2000$.

In table 1 the number of groups of order $1024$ is given, it is $49 487 365 422$. Hence ~99.2% of all groups of order at most $2000$ have order $1024$.

EDIT (2022): See the other answer for a recent correction to the precise number.

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Burrell notices here, that there is a mistake in the original calculation. Actually a recent enumeration of groups of order $2^{10}$ shows, that there are $$ 49487367289 $$ groups of this order, and not $49487365422$. The data has been corrected recently also on OEIS. Clearly this does not matter for the present question. Still more than $99\%$ of all groups of order $n\le 2000$ have order $1024$.

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