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<p>For any nonnegative integer <i>n</i>, a family of groups, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathcal {D}_n \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, was introduced by Bianchi et al., as the collection of all finite groups with exactly <i>n</i> conjugacy classes of nontrivial, non-selfnormalizing subgroups. It was conjectured that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {D}_5\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">D</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation> consists of solvable groups with derived length at most 3. In this note we verify their conjecture.</p>

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Groups with 5 Nontrivial Conjugacy Classes of Non-Selfnormalizing Subgroups are Solvable

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • Maria Loukaki

摘要

For any nonnegative integer n, a family of groups, denoted by \( \mathcal {D}_n \) D n , was introduced by Bianchi et al., as the collection of all finite groups with exactly n conjugacy classes of nontrivial, non-selfnormalizing subgroups. It was conjectured that \(\mathcal {D}_5\) D 5 consists of solvable groups with derived length at most 3. In this note we verify their conjecture.