Naturalness of subgroup rank versus sectional p-rank

Determine whether the largest rank r_p(G) of an elementary abelian p-subgroup is also a natural invariant for the group-cohomological applications in which the sectional p-rank s_p(G) controls the dimension and complexity of group cohomology.

Background

The appendix proves an inequality relating sectional p-rank, defined through elementary abelian subquotients, to subgroup rank, defined through elementary abelian subgroups. The authors explain that sectional p-rank is already used in group cohomology to control dimension and complexity.

They explicitly leave unresolved whether the subgroup-only invariant r_p(G) can play the same natural role in that setting. This is a narrower structural question about the relevance of quotient information versus actual subgroup information.

References

The sectional $p$-rank of a group has applications outside of this work. It is an important invariant in group cohomology, where it controls the dimension and complexity of group cohomology; see for example . \cref{lem:sectional-p-rank} allows the results of the aforementioned papers to be stated in terms of existence of subgroups only; it is not clear to us whether $r_p(G)$ is also a natural invariant in that setting.

— Abelian structure in approximate groups and Alon's conjecture on Ramsey Cayley graphs  (2512.15125 - Schildkraut, 17 Dec 2025) in Appendix A, Section A.1 (Understanding the sectional p-rank)