Classification of saturated nonsolvable groups

Classify the nonsolvable finite groups Γ for which N(Γ)=μ(Γ), equivalently for which the associated Galois class has equal Natarajan and DS dimensions.

Background

The paper proves saturation for every solvable group and also for the nonsolvable group PSL(2,8), while A_5, S_5, and PSL(2,7) are nonsaturated. These examples show that solvability is sufficient but not necessary for equality.

A general classification of nonsolvable groups with N(Γ)=μ(Γ) is left unresolved.

References

Finally, which non-solvable groups are saturated? $\mathrm{PSL}(2,8)$ is (\cref{prop:psl28}); $A_5$ and $S_5$ are not (\cref{thm:solvable}(b)), and neither is $\mathrm{PSL}(2,7)$ (by exhaustive computer search, \cref{tab:cosetcube}).

— 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