Residual separability of subgroups in free products with amalgamated subgroup of finite index
A. A. Kryazheva Ivanovo State University, Ivanovo, Russia
Abstract:
Let
$P$ be the free product of groups
$A$ and
$B$ with amalgamated subgroup
$H$, where
$H$ is a proper subgroup of finite index in
$A$ and
$B$. We assume that the groups
$A$ and
$B$ satisfy a nontrivial identity and for each natural
$n$ the number of all subgroups of index
$n$ in
$A$ and
$B$ is finite. We prove that all cyclic subgroups in
$P$ are residually separable if and only if
$P$ is residually finite and all cyclic subgroups in
$H$ are residually separable; and all finitely generated subgroups in
$P$ are residually separable if and only if
$P$ is residually finite and all subgroups that are the intersections of
$H$ with finitely generated subgroups of
$P$ are finitely separable in
$H$.
Keywords:
residually separable subgroup, residually finite group, free product, split extension.
UDC:
512.543 Received: 19.07.2018
Revised: 19.07.2018
Accepted: 17.10.2018
DOI:
10.33048/smzh.2019.60.212
Fulltext:
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