Abstract <p>Suppose that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}\)</EquationSource>  </InlineEquation> is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource>  </InlineEquation> is the free product of residually <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}\)</EquationSource>  </InlineEquation>-groups <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(A_{i}\)</EquationSource>  </InlineEquation> (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(i\in\mathcal{I}\)</EquationSource>  </InlineEquation>), and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation> is a subgroup of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource>  </InlineEquation> satisfying a non-trivial identity. We prove a criterion for the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}\)</EquationSource>  </InlineEquation>-separability of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation> in <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource>  </InlineEquation>. It follows from this criterion that, if <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{\mathcal{V}_{j}|j\in\mathcal{J}\}\)</EquationSource>  </InlineEquation> is a family of group varieties, each <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{V}_{j}\)</EquationSource>  </InlineEquation> (<InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(j\in\mathcal{J}\)</EquationSource>  </InlineEquation>) is distinct from the variety of all groups, and <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq14.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{V}=\bigcup_{j\in\mathcal{J}}\mathcal{V}_{j}\)</EquationSource>  </InlineEquation>, then one can give a description of <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{C}\)</EquationSource>  </InlineEquation>-separable <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{V}\)</EquationSource>  </InlineEquation>-subgroups of <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource>  </InlineEquation> provided such a description is known for every group <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(A_{i}\)</EquationSource>  </InlineEquation> (<InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8442_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(i\in\mathcal{I}\)</EquationSource>  </InlineEquation>).</p>

On the Subgroup Separability of the Free Product of Groups

期刊论文
发表日期：

2025年7月1日

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E. V. Sokolov

摘要

Abstract

Suppose that \(\mathcal{C}\) is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), \(G\) is the free product of residually \(\mathcal{C}\) -groups \(A_{i}\) ( \(i\in\mathcal{I}\) ), and \(H\) is a subgroup of \(G\) satisfying a non-trivial identity. We prove a criterion for the \(\mathcal{C}\) -separability of \(H\) in \(G\) . It follows from this criterion that, if \(\{\mathcal{V}_{j}|j\in\mathcal{J}\}\) is a family of group varieties, each \(\mathcal{V}_{j}\) ( \(j\in\mathcal{J}\) ) is distinct from the variety of all groups, and \(\mathcal{V}=\bigcup_{j\in\mathcal{J}}\mathcal{V}_{j}\) , then one can give a description of \(\mathcal{C}\) -separable \(\mathcal{V}\) -subgroups of \(G\) provided such a description is known for every group \(A_{i}\) ( \(i\in\mathcal{I}\) ).