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 ?



Asymptotic density of finite abelian and solvable groups
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.


