On the supersolubility of a finite group factorized into pairwise permutable seminormal subgroups
Tom 164 / 2021
Colloquium Mathematicum 164 (2021), 175-183
MSC: 20D10, 20D40.
DOI: 10.4064/cm8091-12-2019
Opublikowany online: 4 September 2020
Streszczenie
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.
