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<p>The non-inner automorphism conjecture (NIAC) and the divisibility problem (DP) are two famous problems in the study of finite <i>p</i>-groups. We observe that the verification of NIAC can be reduced to purely non-abelian finite <i>p</i>-groups. In connecting NIAC with DP, as a consequence of our results obtained on NIAC, we provide a short and cohomology-free proof of a theorem of Yadav, which states that if <i>G</i> is a finite <i>p</i>-group such that (<i>G</i>,&#xa0;<i>Z</i>(<i>G</i>)) is a Camina pair, then |<i>G</i>| divides <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2095_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(|{{\,\mathrm{\!Aut}\,}}(G)|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mrow> <mspace width="0.166667em" /> <mrow> <mspace width="-0.166667em" /> <mi mathvariant="normal">Aut</mi> </mrow> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Automorphisms of finite p-groups

  • 期刊论文

  • 发表日期：

    2025年2月20日

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  • Hemant Kalra,
  • Deepak Gumber

摘要

The non-inner automorphism conjecture (NIAC) and the divisibility problem (DP) are two famous problems in the study of finite p-groups. We observe that the verification of NIAC can be reduced to purely non-abelian finite p-groups. In connecting NIAC with DP, as a consequence of our results obtained on NIAC, we provide a short and cohomology-free proof of a theorem of Yadav, which states that if G is a finite p-group such that (G, Z(G)) is a Camina pair, then |G| divides \(|{{\,\mathrm{\!Aut}\,}}(G)|\) | Aut ( G ) | .