<p>Consider some class&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ C $</EquationSource> </InlineEquation>of groups that is closed under subgroups, quotients,and unrestricted wreath products, and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ G $</EquationSource> </InlineEquation> be the generalized free productof groups&#xa0;<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ A $</EquationSource> </InlineEquation> and&#xa0;<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ B $</EquationSource> </InlineEquation>with&#xa0;an&#xa0;amalgamated subgroup&#xa0;<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$ H $</EquationSource> </InlineEquation>,which is a&#xa0;normal and proper subgroup of the free factors.Suppose thatthe subgroup of the automorphism group of&#xa0;<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$ H $</EquationSource> </InlineEquation>consisting of the restrictions to&#xa0;<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$ H $</EquationSource> </InlineEquation> of all inner automorphisms of&#xa0;<InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ G $</EquationSource> </InlineEquation>is finite,or abelian,or generated by the restrictions of inner automorphisms of one of the free factors.In&#xa0;this article we describe the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ C $</EquationSource> </InlineEquation>-separable finitely generated abelian subgroupsof&#xa0;<InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ G $</EquationSource> </InlineEquation>on assuming thatthe latter is a&#xa0;residually&#xa0;<InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ C $</EquationSource> </InlineEquation>-group.A&#xa0;criterion for the <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$ C $</EquationSource> </InlineEquation>-residuality of&#xa0;<InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1533_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$ G $</EquationSource> </InlineEquation>is available.</p>

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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

摘要

Consider some class  $ C $ of groups that is closed under subgroups, quotients,and unrestricted wreath products, and let $ G $ be the generalized free productof groups  $ A $ and  $ B $ with an amalgamated subgroup  $ H $ ,which is a normal and proper subgroup of the free factors.Suppose thatthe 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 subgroupsof  $ G $ on assuming thatthe latter is a residually  $ C $ -group.A criterion for the $ C $ -residuality of  $ G $ is available.