The Geometric Isomorphism Test

The modular isomorphism problem asks: if two groups have isomorphic group rings over a given ring, must the groups themselves be isomorphic? For group rings over fields of characteristic p, where p divides the group order, the question has resisted general solution for decades.

Margolis and Cohrs approach it geometrically. They develop a procedure based on computational algebraic geometry — treating the isomorphism question as a system of polynomial equations whose solvability can be determined by Gröbner basis methods and variety analysis.

Applied to groups of order 64 (= 2⁶) over commutative unital rings where 2 is not invertible, they establish: if RG ≅ RH as unital algebras, then G ≅ H. The group ring determines the group.

The geometric method works because it transforms an abstract algebraic question (are these groups isomorphic?) into a concrete geometric one (does this algebraic variety have points?). The variety encodes all possible isomorphisms between the group rings, and its emptiness or non-emptiness resolves the isomorphism question. For groups of order 64, the varieties are computationally accessible — large enough to be non-trivial, small enough for Gröbner basis algorithms to terminate.

Order 64 is the frontier. Earlier orders were resolved by hand or by simpler invariants. Higher orders remain open. The geometric approach offers a systematic path forward — but the computational cost of variety analysis grows with group order, and whether the method scales is itself an open question.