Let G be a finite group. We denote by ν(G) the probability that two randomly chosen elements of G generate a nilpotent subgroup. In this paper, we characterize the structure of finite groups G with lower bounds <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2510_Article_IEq1.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">${1 \over p}, \, {{p^{2}+8} \over {9p^{2}}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow> <mfrac> <mrow> <msup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> </mrow> <mrow> <mn>9</mn> <msup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2510_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${p+3} \over {4p}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mi>p</mi> </mrow> </mfrac> </math></EquationSource> </InlineEquation> on ν(G), where p is a prime divisor of ∣G∣.

The Nilpotent Probability of Finite Groups

期刊论文
发表日期：

2025年4月15日

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Huaquan Wei,
Xuanyou Hou,
Changman Sun,
Xixi Diao,
Hui Wu,
Liying Yang

摘要

Let G be a finite group. We denote by ν(G) the probability that two randomly chosen elements of G generate a nilpotent subgroup. In this paper, we characterize the structure of finite groups G with lower bounds ${1 \over p}, \, {{p^{2}+8} \over {9p^{2}}}$ $\frac{1}{p}, \frac{p^{2} + 8}{9 p^{2}}$ and ${p+3} \over {4p}$ $\frac{p + 3}{4 p}$ on ν(G), where p is a prime divisor of ∣G∣.