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<p>In this work we prove that a locally graded group whose proper subgroups are (locally nilpotent)-by-Chernikov is itself (locally nilpotent)-by-Chernikov. We prove also that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\varvec{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">G</mi> </mrow> </math></EquationSource> </InlineEquation> is an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak {X}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">X</mi> </math></EquationSource> </InlineEquation> -group of infinite rank whose proper subgroups of infinite rank are (locally nilpotent)-by-Chernikov (respectively, Baer-by-Chernikov), then so is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\varvec{G}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">G</mi> </mrow> </math></EquationSource> </InlineEquation>; where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathfrak {X}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">X</mi> </math></EquationSource> </InlineEquation> is the class defined by N. S. Chernikov as the closure of the class of periodic locally graded groups by the closure operations <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( \varvec{\acute{P}}, \varvec{\grave{P}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">P</mi> <mo mathvariant="bold">´</mo> </mover> </mrow> <mo>,</mo> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">P</mi> <mo mathvariant="bold">`</mo> </mover> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( \varvec{L}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">L</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Locally graded groups with many (locally nilpotent)-by-Chernikov subgroups

  • 期刊论文

  • 发表日期：

    2026年7月28日

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  • Sevgi Atlihan,
  • Abdelhafid Badis,
  • Amel Dilmi,
  • Nadir Trabelsi

摘要

In this work we prove that a locally graded group whose proper subgroups are (locally nilpotent)-by-Chernikov is itself (locally nilpotent)-by-Chernikov. We prove also that if \({\varvec{G}}\) G is an \(\mathfrak {X}\) X -group of infinite rank whose proper subgroups of infinite rank are (locally nilpotent)-by-Chernikov (respectively, Baer-by-Chernikov), then so is \({\varvec{G}}\) G ; where \(\mathfrak {X}\) X is the class defined by N. S. Chernikov as the closure of the class of periodic locally graded groups by the closure operations \( \varvec{\acute{P}}, \varvec{\grave{P}}\) P ´ , P ` and \( \varvec{L}\) L .