English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Suppose that $\mathcal{C}$ is a class of groups consisting only of periodicgroups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numberseach 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 $x \in X \setminus Y$, thereexists 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 $x \in X$, $q \in \mathfrak{P}(\mathcal{C})^{\prime}$, theinclusion $x^{q} \in Y$ 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 nilpotentgroup $N$ to have the property $\mathcal{C}\mbox{-}\mathfrak{Sep}$ provided$\mathcal{C}$ is a root class (i.e., it contains non-trivial groups and isclosed 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 anisomorphic copy of $U$ for each $v \in V$). We also prove that if $N$ istorsion-free, then the indicated condition is necessary for this group to have$\mathcal{C}\mbox{-}\mathfrak{Sep}$.

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