Does someone know of any works done regarding the density of rank k groups in the collection of groups of order $p^n$ ? i.e.- do we know that almost all p-groups are of rank 3 for example?


Thanks in advance for any suggestions



Question regarding rank k p-groups
In the paper


Higman, Graham,
Enumerating $p$-groups. I. Inequalities.
Proc. London Math. Soc. (3) 10 1960 24–30, 


it is proved that the number of isomorphism-classes of groups of order $p^n$ is $p^{An^3}$, where


$2/27−o(n)≤A≤2/15+o(n)$.


The upper bound was improved by Sims in


Sims, Charles C.,
Enumerating $p$-groups.
Proc. London Math. Soc. (3) 15 1965 151–166.


where it is shown that $A=2/27+O(n^{−1/3})$.


Higman established the lower bound by considering only $p$-groups of class 2, in which $G/\Phi(G)$ and $\Phi(G)$ are elementary abelian of orders about $p^{2n/3}$ and $p^{n/3}$ respectively.


Although this does not prove that almost all groups of order $p^n$ have rank about $2n/3$, it suggests that that might be the case.


