Factorizations of Finite Groups into r-Soluble Subgroups with Given Embeddings - Knyagina, V. N.; Tyutyanov, V. N. (Ukrains’kyi Matematychnyi Zhurnal, 2014)
Let X be a subset of the set of positive integers. A subgroup H of a group G is called X-subnormal in G if there exists a chain of subgroups H = H_0 ⊆ H_1 ⊆ … ⊆ H_n = G such that |H_i : H_i-1| ∈ X for all i. We study the solubility and r -solubility of a finite group G = AB with some restrictions imposed on the subgroups A and B and on the set X .