On k-submodular subgroups of finite groups

Let n be a natural number. A subgroup H of a group G will be called n-modularly embedded in G if either H is normal in G or H= CoreG(H), $|G:H| =p$ and |G/CoreG(H)|=pqⁿ, qⁿ divides $p-1$ for some primes p and q. Let k be a fixed natural number. A subgroup H of a group G will be called k-submodular in G if there exists a chain H = H₀ ≤ H₁ ≤ ⋯ ≤ Hₘ₋₁ ≤ Hₘ = G of subgroups such that Hᵢ₋₁ is n-modularly embedded in Hᵢ for some natural n≤ k and every i = 1, … , m. In this article, properties of k-submodular subgroups and classes of groups with given systems of k-submodular subgroups are obtained.