Asymptotic density of finite abelian and solvable groups 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 ?
 A: 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.
A: (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.
