Finite groups for which the spectral analogue fails

Determine which finite groups admit a Rota–Baxter operator B such that, for every N in the positive integers and every m with 0 ≤ m ≤ N, the composite operator B^m composed with the N−m power of the dual operator B̃ is nontrivial.

Background

For finite-dimensional algebras, the spectrum of a Rota–Baxter operator of weight λ is contained in {0, −λ}, which yields a finite nilpotency-type invariant rbλ(A). The paper asks when the corresponding group-theoretic phenomenon fails for finite groups, using the dual operator B̃ defined from a group Rota–Baxter operator B.

The unresolved problem is to characterize all finite groups for which there exists a Rota–Baxter operator whose iterated compositions with its dual never become trivial, regardless of the prescribed exponent N. The paper studies this phenomenon for the finite Heisenberg group but does not provide a classification of all finite groups with this property.

References

For which finite groups will the analogue of this property fail to hold? Equivalently, for which finite groups does there exist a Rota---Baxter operator B for any N \in \mathbb{N} such that Bm \widetilde{B}{N-m} \neq { e } for all m \leq N?

— Rota---Baxter operators on extraspecial groups  (2608.28291 - Savelyev, 28 Aug 2026) in Section 5, subsection “Analogue of the Spectral Property”