Let G be a finite non-Dedekind p-group. We denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the number of orders of abelian non-normal subgroups of G. In this paper, we bound the order and the nilpotency class of G based on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma (G)=t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|G|\le p^{(4t-2)(8t-3)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msup> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mi>t</mi> <mo>-</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and that the nilpotency class of G is at most <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$16t-9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>16</mn> <mi>t</mi> <mo>-</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation>, except for the case where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> is cyclic.

Finite p-Groups Whose Abelian Non-normal Subgroups Have Few Orders

期刊论文
发表日期：

2025年12月18日

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Lijuan He,
Adolfo Ballester-Bolinches,
Heng Lv

摘要

Let G be a finite non-Dedekind p-group. We denote by $\gamma (G)$ $γ (G)$ the number of orders of abelian non-normal subgroups of G. In this paper, we bound the order and the nilpotency class of G based on $\gamma (G)$ $γ (G)$ . Let $\gamma (G)=t$ $γ (G) = t$ . We show that $|G|\le p^{(4t-2)(8t-3)}$ $| G | \leq p^{(4 t - 2) (8 t - 3)}$ and that the nilpotency class of G is at most $16t-9$ $16 t - 9$ , except for the case where $p=2$ $p = 2$ or $G'$ $G^{'}$ is cyclic.