Neumann–Praeger covering-subgroup conjecture

Establish whether there exists a function f: N -> N such that, for every finite group A with a normal subgroup H of index n and every A-covering subgroup U of H satisfying H = ⋃_{a∈A} U^a, the index |H:U| is at most f(n).

Background

The paper explains that conjectures concerning Kronecker classes of field extensions can be reformulated as covering problems for finite groups. A subgroup U of H is called an A-covering subgroup when the union of its A-conjugates equals H.

The Neumann–Praeger conjecture asks for a bound on |H:U| depending only on the index n = |A:H|. The conjecture is known for n = 1 and n = 2; the paper proves it for n = 3 by applying its clique result for derangement graphs.

References

Conjecture 1.2 (Neumann-Praeger). Let A be a finite group with a normal subgroup H. A subgroup U of H is called an A-covering subgroup of H if H = UaEA Uª. Is there a function f : N -> N such that whenever U < H ≤ A, where A is a finite group, H is a normal subgroup of A of index n, and U is an A-covering subgroup of H, the index | H : U| ≤ f(n)?

— Kronecker classes and cliques in derangement graphs  (2502.01287 - Cazzola et al., 3 Feb 2025) in Conjecture 1.2, Section 1