Remove the logarithms from the asymptotic prevalence of p-groups

Establish whether the number of finite p-groups of order at most n is asymptotically equivalent, without taking logarithms, to the number of all finite groups of order at most n.

Background

The paper recalls Pyber’s result that p-groups are log-asymptotically prevalent among all finite groups: the ratio of the logarithms of the two counting functions tends to one. It then notes that the stronger statement obtained by removing the logarithms is a longstanding conjecture. This conjecture would strengthen the enumerative basis for viewing p-groups, and especially 2-groups, as representative hard cases of finite-group isomorphism.

References

It is a well-known, long open conjecture that this is true without the logs; see, e.g., .

— Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2  (2610.00874 - Grochow et al., 1 Oct 2026) in Section 1, subsection “The significance of 2-groups of Frattini class 2”