Moretó’s subgroup-chain conjecture for p-divisible irreducible characters

Determine whether, for every finite group G, prime p, and Sylow p-subgroup P of G, the length of every subgroup chain between the normalizer N_G(P) and G is at most the number of irreducible characters of G whose degrees are divisible by p.

Background

The paper presents Moretó’s conjecture as an extension of the Itô–Michler theorem to finite groups with nonabelian or nonnormal Sylow subgroups. If G has n irreducible characters whose degrees are divisible by p, the conjecture asserts that every subgroup chain from N_G(P) to G has length at most n.

The paper subsequently constructs a counterexample with |Irr_p(G)| = 2 and l_p(G) = 3, thereby disproving the conjecture in general. It also notes that the conjecture had previously been proved when |Irr_p(G)| = 1 and states that affirmative results are obtained in a forthcoming manuscript for certain solvable groups.