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Abstract

A subgroup $A$ of a group $G$ is called seminormal in $G$ if there exists a subgroup $B$ such that $G=AB$ and $AX$ is a subgroup of $G$ for every subgroup $X$ of $B$. We study groups $G = G_1\ldots G_n$ with pairwise permutable subgroups $G_1, \ldots , G_n$ such that $G_i$ and $G_j$ are seminormal in $G_iG_j$ for any $i, j\in \{1,\ldots , n\}$, $i\neq j$. In particular, we prove that $G$ is supersoluble in the following cases: $G_1$ is supersoluble and $G_i$ is nilpotent for every $i\geq 2$; $G_i$ is supersoluble for any $i$ and $G^\prime $ is nilpotent.