Two-thirds lower bound for coset-cube numbers

Determine whether every finite group Γ satisfies N(Γ) ≥ ⌈2μ(Γ)/3⌉.

Background

The paper explains that every finite group can be realized as a Galois group and that, on a suitable finite faithful domain, the DS and Natarajan dimensions become μ(Γ) and N(Γ), respectively. Thus a universal lower bound on N(Γ) in terms of μ(Γ) would strengthen the half-factor conjecture in the Galois setting.

The group A_5 is identified as an obstruction to improving the constant 2/3, since its values are μ(A_5)=3 and N(A_5)=2.