On a conjecture related to the Davenport constant

For a finite group G, D (G) is defined as the least positive integer k such that for every sequence S=g₁ g₂ gₖ of length k over G, there exist 1 i₁ < i₂ < < iₘ k such that ₉=₁^m g₈_ (₉) =1 holds for = id, identity element of Sₘ. For a finite abelian group, this group invariant, known as the Davenport constant, is crucial in the theory of non-unique factorization domains. The precise value of this invariant, even for a finite abelian group of rank greater than 2, is not known yet. In 1977, Olson and White first worked with this invariant for finite non-abelian groups. After that in 2004, Dimitrov dealt with it, where he proved that D (G) L (G) for a finite p-group G, where p is a prime and L (G) is the Loewy length of FₚG. He conjectured that equality holds for all finite p-groups. In this article, we compute D (G) for a certain subclass of 2-generated finite p-groups of nilpotency class two and show that the conjecture is true by determining the precise value of the Loewy length of FₚG. We also evaluate D (G) for finite dicyclic, semi-dihedral and some other groups.