Scholarlink

Abstract <p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}\)</EquationSource> <!--RusMath2570032Sokolov-m1--> </InlineEquation> be a nonempty set of primes. We prove that any <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}\)</EquationSource> <!--RusMath2570032Sokolov-m2--> </InlineEquation>-bounded nilpotent group is <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}\)</EquationSource> <!--RusMath2570032Sokolov-m3--> </InlineEquation>-potent and the tree product <i>T</i> of a finite number of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}\)</EquationSource> <!--RusMath2570032Sokolov-m4--> </InlineEquation>-bounded nilpotent groups with proper locally cyclic edge subgroups is residually a finite <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}\)</EquationSource> <!--RusMath2570032Sokolov-m5--> </InlineEquation>-group iff any vertex group of <i>T</i> has no <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}{\kern 1pt} '\)</EquationSource> <!--RusMath2570032Sokolov-m6--> </InlineEquation>-torsion and any edge subgroup of <i>T</i> is <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{P}{\kern 1pt} '\)</EquationSource> <!--RusMath2570032Sokolov-m7--> </InlineEquation>-isolated in the vertex group containing it. We prove also that the tree product of a finite number of groups with locally cyclic edge subgroups is residually a finite <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> <!--RusMath2570032Sokolov-m8--> </InlineEquation>-group if all its vertex groups have this property and any edge subgroup is separable in the corresponding vertex group by the class of finite <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10488_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\)</EquationSource> <!--RusMath2570032Sokolov-m9--> </InlineEquation>-groups.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Certain Residual Properties of Bounded Nilpotent Groups and Their Tree Products

  • 期刊论文

  • 发表日期：

    2025年4月1日

查看全文
  • E. V. Sokolov

摘要

Abstract

Let \(\mathfrak{P}\) be a nonempty set of primes. We prove that any \(\mathfrak{P}\) -bounded nilpotent group is \(\mathfrak{P}\) -potent and the tree product T of a finite number of \(\mathfrak{P}\) -bounded nilpotent groups with proper locally cyclic edge subgroups is residually a finite \(\mathfrak{P}\) -group iff any vertex group of T has no \(\mathfrak{P}{\kern 1pt} '\) -torsion and any edge subgroup of T is \(\mathfrak{P}{\kern 1pt} '\) -isolated in the vertex group containing it. We prove also that the tree product of a finite number of groups with locally cyclic edge subgroups is residually a finite \(p\) -group if all its vertex groups have this property and any edge subgroup is separable in the corresponding vertex group by the class of finite \(p\) -groups.