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On the separability of abelian subgroups of the generalized free product of two groups with normal amalgamated subgroup
D. R. Baranov,
E. V. Sokolov Ivanovo State University, Ivanovo, Russia
Abstract:
Consider some class
$C$ of groups that is closed under subgroups, quotients, and unrestricted wreath products, and let
$G$ be the generalized free product of groups
$A$ and
$B$ with an amalgamated subgroup
$H$, which is a normal and proper subgroup of the free factors. Suppose that the subgroup of the automorphism group of
$H$ consisting of the restrictions to
$H$ of all inner automorphisms of
$G$ is finite, or abelian, or generated by the restrictions of inner automorphisms of one of the free factors. In this article we describe the
$C$-separable finitely generated abelian subgroups of
$G$ on assuming that the latter is a residually
$C$-group. A criterion for the
$C$-residuality of
$G$ is available.
Keywords:
separability of abelian subgroups, root class residuality, generalized free product.
UDC:
512.543
MSC: 20E26,
20E06 Received: 26.11.2024
Revised: 26.11.2024
Accepted: 25.02.2025
DOI:
10.33048/smzh.2025.66.203
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