On finite factorized groups with $\mathbb TX$-subnormal subgroups

Abstract

UDC 512.542

Let $\mathbb T$ be a subset of the set of all natural numbers satisfying the condition $$\begin{gather}\text{if}\,\, t\in\mathbb{T},\,\, \text{then}\,\, \mathbb{T}\,\, \text{contains all natural divisors of}\quad t.\tag{A}\end{gather}$$ Recall that a subgroup $H$ is called a {\it $\mathbb T$-subnormal} in $G$ if either $H=G,$ or there is a chain of subgroups $H=H_0 \le H_1 \le \ldots \le H_n = G$ such that $|H_i\colon H_{i-1}| \in \mathbb T$ for all $i.$ Let $X$ be a normal subgroup of a group $G$ and let $\Bbb T$ be a set of natural numbers satisfying condition (A). We introduce the following definition: A subgroup $H$ of the group $G$ is called a {\it $\mathbb TX$-subnormal} subgroup if $H$ is $\mathbb T$-subnormal in $HX.$ Moreover, we study factorizable groups $G = AB$ with $\mathbb TX$-subnormal factors $A$ and $B$. Under certain additional restrictions imposed on $A,$ $B,$ $\mathbb T,$ and $X,$ we obtain new sufficient conditions for the partial solubility and supersolubility of the analyzed groups $G$.