Character-degree invariants for finite simple groups

Determine whether the system of invariants consisting of the order n(G) and the indicator tuple δ^{cd}(G) for the set of complex irreducible character degrees is full for the class of finite simple groups.

Background

The paper establishes that the order together with the corresponding indicator tuple for element orders and, separately, conjugacy class sizes, distinguishes finite simple groups. It then asks whether the analogous system based on the set of irreducible character degrees does the same.

The question is weaker than Huppert's conjecture: a positive answer would establish recognition using these invariants, but would not by itself imply the stronger structural conclusion in Huppert's conjecture.

References

Though we do not know whether the system of invariants $(n(G),\delta{cd}(G))$ is full for all simple groups, it follows from that $\delta{cd*}(G)$ is.

— Arithmetic invariants for finite simple and related groups  (2608.12783 - Vasil'ev, 13 Aug 2026) in Problem 1.?, Section 2, after Theorem 2.3

Though we do not know whether the system of invariants $(n(G),\delta{cd}(G))$ is full for the simple groups, we also do not know the answer to the following question.

— Arithmetic invariants for finite simple and related groups  (2608.12783 - Vasil'ev, 13 Aug 2026) in Problem \ref{prob:CdRel}, Section 3, character-degree case