Geodesic
Оn direct decompositions of $n$-multiply $\omega$-saturated formations
Problemy fiziki, matematiki i tehniki, no. 1 (2011), pp. 48-51.

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All groups considered are finite. Let $\{\mathfrak{F}_i \mid i\in I\}$ be a set of non-empty subclasses of a class of groups $\mathfrak{F}$ such that $\mathfrak{F}_i \cap \mathfrak{F}_j = (1)$ for all distinct $i, j \in I$. We write $\mathfrak{F}=\bigoplus_{i\in I}\mathfrak{F}_i$ to denote the collection of all groups of the form $А_1\times \dots \times А_t$, where $A_1 \in \mathfrak{F}_{i_1},\dots,A_t \in \mathfrak{F}_{i_1}$ for some $i_1,\dots, i_t \in I$. We proved the following theorem. Theorem. Let $\mathfrak{F}=\bigoplus_{i \in I} \mathfrak{F}_i$ where $\mathfrak{F}_i$ is a formation. Then $\mathfrak{F}$ is $n$-multiply ($n\ge 1$)$\omega$-saturated formation if and only if $\mathfrak{F}_i$ is $n$-multiply $\omega$-saturated for all $i \in I$.
Keywords: formation of finite groups, complemented subformation, direct decomposition of a class of groups, $\omega$-local satellite, $n$-multiply $\omega$-saturated formation. 
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N. N. Vorob'ev; A. P. Mekhovich. Оn direct decompositions of $n$-multiply $\omega$-saturated formations. Problemy fiziki, matematiki i tehniki, no. 1 (2011), pp. 48-51. http://geodesic.mathdoc.fr/item/PFMT_2011_1_a6/

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