تقرير
On the separability of subgroups of nilpotent groups by root classes of groups
| العنوان: | On the separability of subgroups of nilpotent groups by root classes of groups |
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| المؤلفون: | Sokolov, E. V. |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Group Theory, 20E26, 20F18 (Primary) 20E06, 20E08 (Secondary) |
| الوصف: | Suppose that $\mathcal{C}$ is a class of groups consisting only of periodic groups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numbers each of which does not divide the order of any element of a $\mathcal{C}$-group. A subgroup $Y$ of a group $X$ is called a) $\mathcal{C}$-separable in this group if, for each $x \in X \setminus Y$, there exists a homomorphism $\sigma$ of $X$ onto a group from $\mathcal{C}$ such that $x\sigma \notin Y\sigma$; b) $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated in $X$ if, for any $x \in X$, $q \in \mathfrak{P}(\mathcal{C})^{\prime}$, the inclusion $x^{q} \in Y$ implies that $x \in Y$. It is easy to see that if $Y$ is $\mathcal{C}$-separable in $X$, then it is $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated in this group. Let us say that $X$ has the property $\mathcal{C}\mbox{-}\mathfrak{Sep}$ if all its $\mathfrak{P}(\mathcal{C})^{\prime}$-isolated subgroups are $\mathcal{C}$-separable. We find a condition that is sufficient for a nilpotent group $N$ to have the property $\mathcal{C}\mbox{-}\mathfrak{Sep}$ provided $\mathcal{C}$ is a root class (i.e., it contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{v \in V}U_{v}$, where $U, V \in \mathcal{C}$ and $U_{v}$ is an isomorphic copy of $U$ for each $v \in V$). We also prove that if $N$ is torsion-free, then the indicated condition is necessary for this group to have $\mathcal{C}\mbox{-}\mathfrak{Sep}$. Comment: 18 pages; the English version of the previously published Russian original |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2202.01378 |
| رقم الأكسشن: | edsarx.2202.01378 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |