Question

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

Answer 1

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.