arXiv 2202.013782022-02-04math.GR

关于幂零群的子群的根类可分性

On the separability of subgroups of nilpotent groups by root classes of groups

E. V. Sokolov

备注：: 18 pages; the English version of the previously published Russian original

摘要：

设$\mathcal{C}$是一类只由周期群组成的群，而$\mathfrak{P}(\mathcal{C})^{\prime}$是素数的集合，其中每个素数都不整除$\mathcal{C}$ -群的任何元素的阶。群$X$的子群$Y$称为a）$\mathcal{C}$ -在这个群中可分离的，如果对于每个$x \in X \setminus Y$，存在$X$到来自$\mathcal{C}$的群上的同态$σ$，使得$xσ\notin Yσ$ ; b）$\mathfrak{P}(\mathcal{C})^{\prime}$ -在$X$中孤立的，如果对于任何$x \in X$，$q \in \mathfrak{P}(\mathcal{C})^{\prime}$，包含$x^{q} \in Y$暗示$x \in Y$。很容易看出，如果$Y$在$X$中是$\mathcal{C}$ -可分离的，那么它在该组中是$\mathfrak{P}(\mathcal{C})^{\prime}$ -分离的。假设$X$具有性质$\mathcal{C}\mbox{-}\mathfrak{Sep}$，如果它的所有$\mathfrak{P}(\mathcal{C})^{\prime}$ -孤立子群都是$\mathcal{C}$ -可分的。我们找到了一个条件，该条件足以使幂零群$N$具有性质$\mathcal{C}\mbox{-}\mathfrak{Sep}$，前提是$\mathcal{C}$是根类（即，它包含非平凡群，并且在取子群、扩张和$\prod_{v \in V}U_{v}$形式的笛卡尔积下是封闭的，其中$U, V \in \mathcal{C}$和$U_{v}$是每个$v \in V$的$U$的同构副本。我们还证明了，如果$N$是扭转自由，那么所示的条件是必要的，这个组有$\mathcal{C}\mbox{-}\mathfrak{Sep}$。

英文摘要：

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 $σ$ of $X$ onto a group from $\mathcal{C}$ such that $xσ\notin Yσ$; 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 $\mathcal{C}\mbox{-}\mathfrak{Sep}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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