Two finite groups can have identical-looking group algebras over certain rings and yet be non-isomorphic as groups. Whether the group algebra determines the group — the modular isomorphism problem — has been open for decades and was recently shown to fail in general for groups of order 512.

This paper resolves the question positively for all groups of order 64: if two groups of order dividing 64 have isomorphic group algebras over any commutative unital ring where 2 is not invertible, the groups themselves are isomorphic. The algebra remembers the group.

The method is computational algebraic geometry. Rather than analyzing the algebra abstractly, the authors construct the space of all possible algebra isomorphisms as a variety and determine algorithmically whether it is nonempty. The geometric approach converts an algebraic question about structural coincidence into a computational question about solution existence.

The boundary this establishes is sharp. Order 64 is safe — the algebra is a faithful fingerprint of the group. At order 512, the fingerprint becomes ambiguous. Somewhere between 64 and 512, the algebra loses enough information to confuse distinct groups.

The lesson is about representation fidelity. A group algebra encodes the group’s multiplication table in a different mathematical language. For small groups, the translation is lossless. For large enough groups, the new language lacks the vocabulary to distinguish everything the original could. The transition from faithful to lossy representation is not gradual — it is absent below a threshold and present above it, with the threshold determined by the interplay of group size and ring arithmetic.