Field-independence of positive modular isomorphism results

Determine whether a positive modular isomorphism result for a fixed finite group over the prime field remains valid over every field of the same characteristic.

Background

The paper discusses the dependence of the modular isomorphism problem on the coefficient field. Although García-Lucas, Margolis, and del Río constructed non-isomorphic finite 2-groups with isomorphic group algebras over every field of characteristic 2, this does not resolve whether a positive result established for a particular group over the prime field must continue to hold over all fields of the same characteristic.

The paper later gives a sufficient descent condition for finite extensions whose degree is coprime to a factorial determined by the number of connected components of an automorphism algebraic group. This provides only a partial result and does not resolve the question for arbitrary coefficient fields.

References

They do not, however, answer whether a positive result for a fixed group over the prime field remains valid over every field of the same characteristic, see .

— Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem  (2609.09525 - Liu, 8 Sep 2026) in Section 1, subsection “Modular isomorphism problem”

Whether $F G\congF H$ implies $G\cong H$ for every field $F$ of characteristic $p$ when $D_3(G)=1$.

— Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem  (2609.09525 - Liu, 8 Sep 2026) in Section 1, subsection “Modular isomorphism problem”; Question 2.10 cited from Margolis–Sakurai