Characterization of the coset-cube number

Determine which group-theoretic invariant equals the coset-cube number N(Γ), and hence equals the Natarajan dimension of the Galois class associated with a finite Galois group Γ.

Background

For a finite Galois group Γ, the paper identifies the Natarajan dimension with N(Γ), the maximum dimension of a family of subgroup/coset pairs whose 2t cells are all nonempty. It proves a lower bound using complemented chief factors, but gives an example, PΓL(2,8), where that chief-factor bound is not exact.

The unresolved issue is to find a general structural description or formula for N(Γ), including how additional coordinates such as those arising from sharp 3-transitivity should be incorporated.

References

Which group invariant is $N(\Gamma)$, and hence the Natarajan dimension of every Galois class?

— Learning a Group Element from Its Action: Galois Correspondences, Optimal ERM, and an Abel-Ruffini Threshold for the Natarajan Dimension  (2610.18154 - Li, 13 Oct 2026) in Section 7, Discussion and open problems, paragraph 2

Is there an invariant of the permutation group that equals $\dimN(H{G}{X})$?

— Learning a Group Element from Its Action: Galois Correspondences, Optimal ERM, and an Abel-Ruffini Threshold for the Natarajan Dimension  (2610.18154 - Li, 13 Oct 2026) in Section 7, Discussion and open problems, paragraph 3

Do the analogues of \cref{thm:monodromy}(c) and~(d) hold for $k$-pseudo-cubes, which govern list learning?

— Learning a Group Element from Its Action: Galois Correspondences, Optimal ERM, and an Abel-Ruffini Threshold for the Natarajan Dimension  (2610.18154 - Li, 13 Oct 2026) in Section 7, Discussion and open problems, paragraph 5