A partition strictly between group isomorphism and simultaneous power equivalence

Problem 6.2 of Fernandes–Qureshi–Reis–Ribas, arXiv:2609.20516v1, asks for a partition P on all finite groups with P₁<P<P₂, P₂<P<P₅ or P₅<P<P₆. The exact wording and definitions are in the literature note. Here P₁ is group isomorphism and P₂ requires one bijection preserving all integer powers simultaneously. The strict order is finer-to-coarser.

The first interval asks whether simultaneous power maps forget group information that can be restored partially without recovering isomorphism. Agreement on abelianity restores one such invariant.

The source supplies power-equivalent abelian and nonabelian groups. A strict intermediate partition additionally needs nonisomorphic groups that the added invariant still identifies. Two nonabelian central extensions with distinct center orders provide this second pair.

Define R(G,H) to be simultaneous power equivalence and agreement on abelianity. Identity, inverse and composition prove equivalence; group isomorphisms preserve both conditions. On F₃⁵ let T(u,v) have product (x,z)(y,w)=(x+y,z+w+u(a b′−b a′)+v(c d′−d c′)). T(0,0) is abelian and T(1,0) is not. T(1,0) and T(1,1) are both nonabelian, with centers of orders 27 and 3. The coordinate bijection preserves every integer power. The two pairs establish both strictnesses. The relation is defined on all finite groups, including the trivial group.

A failure of the single-bijection quantifier, the actual group laws, the center calculations or either strictness pair would break this argument. A cardinality-only relation would not establish the source endpoint.

The native module is D5/S3/FiniteGroups/PowerInvariantRefinement.lean. finite_source_endpoint states equivalence and both strict refinements on a bundle with actual Group and Finite carriers. isomorphism_invariant explicitly transports R in both arguments. The strictness proof uses actual centers and group isomorphisms, rather than numerical proxies. The linked Scribe owns the native declaration binding.

This is the first alternative of the literal disjunctive Problem 6.2. The common refinement, prime-exponent power phenomenon and central extensions are classical ingredients. Neither the other intervals nor another numbered problem is claimed.

Worldwide priority and completeness of the current correction/solution search are unverified. This elementary combination is not claimed as an original group construction. Repository integration and external review are separate from the mathematical statement.