For every finite solvable group G, every element g∈ G such that chi(g)≠ 0 for every irreducible complex character chi∈operatornameIrr(G) belongs to the Fitting subgroup mathbfF(G).

Isaacs–Navarro–Wolf conjecture

Fitting-subgroup containment formulation

For every finite solvable group $G$ , the set $\mathcal{N}(G)$ of non-vanishing elements satisfies $\mathcal{N}(G)\subseteq\mathbf{F}(G)$ , where $\mathcal{N}(G)=\{g\in G:\chi(g)\neq 0\text{ for every }\chi\in\operatorname{Irr}(G)\}$ .

Isaacs, Navarro, and Wolf introduced the conjecture in 1999: every element of a finite solvable group outside its Fitting subgroup should be vanishing, equivalently $\mathcal{N}(G)\subseteq\mathbf{F}(G)$ .

Known results

It was known for solvable groups of odd order.
It was known when all Sylow $2$ -subgroups are abelian.
Earlier bounds placed $\mathcal{N}(G)$ inside $\mathbf{F}_{10}(G)$ , with later progress covering cases involving the first seven Fitting factors.
Related work on monomial characters established restricted variants, but not the original conjecture.

August 24, 2026 claimed proof

The preprint A Proof of the Isaacs–Navarro–Wolf Conjecture claims a proof for finite solvable groups using finite symplectic spaces and Clifford theory. This is a complete-resolution claim, but no independent verification, referee report, or resolution of possible gaps was found.

Current status (as of August 2026): A proof for finite solvable groups is claimed in the August 2026 preprint but remains unverified; no resolution beyond that scope is established.

Isaacs–Navarro–Wolf conjecture

Fitting-subgroup containment formulation

Primary source

Additional references

Progress summary

Solutions 0