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 $X$ .