Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

For every natural number n, let:

  • Gn be the number of distinct group structures with at most n elements;

  • An be the number of distinct abelian group structures wit at most n elements;

  • Sn be the number of distinct solvable group structures with at most n elements.

Question 1: Is there a known limit for the quotient An/Gn ?

Question 2: Is there a known limit for the quotient Sn/Gn ?

share|improve this question

2 Answers 2

up vote 7 down vote accepted

The number of abelian groups of order at most $n$ is $O(n)$, whereas if $n=2^k$, the number of class $2$ nilpotent groups of order $n$ is $2^{(2/27)k^3+O(k^{8/3})}=n^{\Omega(\log^2n)}$ by a result of Sims, hence the answer to question 1 is $0$. It is conjectured that the global asymptotic density of $2$-groups of nilpotent class $2$, and a fortiori of solvable groups, is $1$, but as far as I know, this has not been proved.

share|improve this answer

(Edited following Emil Jerabek's coment below) From results of L. Pyber (and implicitly, C. Sims) it appears likely that $\frac{f(n)}{g(n)} \to 1$ as $n \to \infty,$ where $f(n)$ is the number of isomorphism types of nilpotent groups of order $n$ and $g(n)$ is the number of isomorphism types of all groups of order $n,$ so minor modifications should yield the same answer for question 2 (which is a cumulative version- note also that all nilpotent groups are solvable). Also, the asymptotic behaviour of the number of isomorphism types of Abelian groups of order $n$ and the number of isomorphism types of nilpotent groups of order $n$ are known: both are multiplicative, so it suffices to consider the case of $p$-groups. The number of isomorphism types of Abelian groups of order $p^{k}$ is $p(k),$ the number of partitions of $k,$ which behaves like $e^{c \sqrt{k}}$ for some (known!) constant $c.$ The number of isomorphism types of groups of order $p^{k}$ is asymptotically around $p^{\frac{2k^{3}}{27}}$ (proved by C. Sims and G. Higman). This suggests that the limit of question 1 should be zero, though again you ask for a cumulative version.

share|improve this answer
2  
Pyber’s results give $(\log f(n))/(\log g(n))\to1$. He didn’t prove, but only conjectured, the stronger statement $f(n)/g(n)\to1$. –  Emil Jeřábek Aug 16 at 10:46
    
OK, thanks, I've reworded my post. –  Geoff Robinson Aug 16 at 12:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.