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- W4310829097 abstract "Let $G$ be a $p$-group. We denote by $mathcal{X}_i(G)$ the intersection of all subgroups of $G$ having index $p^i$, for $i leq log_p(|G|)$. In this paper, the newly introduced series ${mathcal{X}_i(G)}_i$ is investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroup $A$ of $G$ intersects each one of the subgroups $mathcal{X}_i(G)$ at $mathcal{X}_i(A)$, then $A$ has a complement in $G$. Conversely if an arbitrary subgroup $H$ of $G$ has a normal complement, then $mathcal{X}_i(H) = mathcal{X}_i(G) cap H$." @default.
- W4310829097 created "2022-12-19" @default.
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- W4310829097 date "2022-12-05" @default.
- W4310829097 modified "2023-10-04" @default.
- W4310829097 title "The $mathcal{X}$-series of a $p$-group and complements of abelian subgroups" @default.
- W4310829097 doi "https://doi.org/10.1007/s00025-023-01927-1" @default.
- W4310829097 hasPublicationYear "2022" @default.
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