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<p>Let <i>G</i> be a finite group and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(h_m(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the harmonic mean of element orders of <i>G</i>. In this short note, we prove that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(h_m(G)&lt;\frac{60}{23}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>&lt;</mo> <mfrac> <mn>60</mn> <mn>23</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>&#xa0;, then <i>G</i> is solvable.</p>

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A criterion for solvability of a finite group by the harmonic mean of element orders

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月2日

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  • Iulia Cătălina Pleşca,
  • Marius Tărnăuceanu

摘要

Let G be a finite group and \(h_m(G)\) h m ( G ) be the harmonic mean of element orders of G. In this short note, we prove that if \(h_m(G)<\frac{60}{23}\) h m ( G ) < 60 23  , then G is solvable.