论文标题
关于nilpotent组的亚组的可分离性
On the separability of subgroups of nilpotent groups by root classes of groups
论文作者
论文摘要
假设$ \ MATHCAL {C} $是仅由周期组组成的一类组和$ \ Mathfrak {p}（\ Mathcal {C}）^{\ Prime} $是一组素数，每个数字都不划分$ \ Mathcal {c} $组的任何元素的顺序。一组$ x $的亚组$ y $称为a）$ \ nathcal {c} $ - 如果在x \ setminus y $中的每个$ x \中，则在此组中可分开，存在$ x $的$ x $的$ x $的$ x $，从$ \ mathcal {c} $ x $xσ\ notin ytin y维; b）$ \ mathfrak {p}（\ mathcal {c}）^{\ prime} $ - 在$ x $中孤立，如果为x $，$ x \ in x $，$ q \ in \ mathfrak {p} {p}（\ mathcal {c}）很容易看出，如果$ y $是$ \ mathcal {c} $ - 在$ x $中可分开，那么它是$ \ mathfrak {p}（\ mathcal {c}）^{\ prime} $ - 在此组中隔离。让我们说，$ x $具有属性$ \ mathcal {c} \ mbox { - } \ mathfrak {sep} $，如果其所有$ \ mathfrak {p}（\ mathcal {c}）^{\ prime}^{\ prime} $ - 孤立的子格鲁普斯是$ \ mathcal as $ \ mathcal calcal {c} $ - c} $ - capable。 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 \在v} u_ {v} $中，其中$ u，v \ in \ mathcal {c} $和$ u_ {v} $是$ u $的同构副本，用于$ u $，每个$ v \ in v $）。我们还证明，如果$ n $不含扭转，则指示条件对于该组具有$ \ mathcal {c} \ mbox { - } \ mathfrak {sep} $。
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 $\mathcal{C}\mbox{-}\mathfrak{Sep}$.
