Given a finite group G of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$p^nm$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mi>n</mi> </msup> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, where p is a prime and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\not \mid m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, we denote by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi _p(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the sum of orders of p-parts of elements in G. In the current note, we prove that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi _p(G)\le \psi _p(C_{p^nm})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>ψ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow> <msup> <mi>p</mi> <mi>n</mi> </msup> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{p^nm}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <msup> <mi>p</mi> <mi>n</mi> </msup> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the cyclic group of order <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2426_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$p^nm$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mi>n</mi> </msup> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, and the equality holds if and only if G is p-nilpotent of a particular type. A generalization of this result is also presented.

A Result on Certain Sums of Element Orders in Finite Groups

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2025年5月20日

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Marius Tărnăuceanu

摘要

Given a finite group G of order $p^nm$ $p^{n} m$ , where p is a prime and $p\not \mid m$ $p ∤ m$ , we denote by $\psi _p(G)$ $ψ_{p} (G)$ the sum of orders of p-parts of elements in G. In the current note, we prove that $\psi _p(G)\le \psi _p(C_{p^nm})$ $ψ_{p} (G) \leq ψ_{p} (C_{p^{n} m})$ , where $C_{p^nm}$ $C_{p^{n} m}$ is the cyclic group of order $p^nm$ $p^{n} m$ , and the equality holds if and only if G is p-nilpotent of a particular type. A generalization of this result is also presented.