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<p>Let <i>p</i> be a prime number. A longstanding conjecture asserts that every finite non-abelian <i>p</i>-group has a non-inner automorphism of order <i>p</i>. In this paper, under some conditions on an odd order finite <i>p</i>-group <i>G</i> with cyclic center, we prove that <i>G</i> exhibits a non-inner automorphism of order <i>p</i>. As a consequence, under certain conditions on a finite <i>p</i>-group <i>G</i> <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2112_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\((p&gt;2),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>&gt;</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3,&#xa0; which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).</p>

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Finite p-groups with cyclic center have non-inner automorphisms of order p

  • 期刊论文

  • 发表日期：

    2025年3月19日

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  • Mandeep Singh,
  • Mahak Sharma

摘要

Let p be a prime number. A longstanding conjecture asserts that every finite non-abelian p-group has a non-inner automorphism of order p. In this paper, under some conditions on an odd order finite p-group G with cyclic center, we prove that G exhibits a non-inner automorphism of order p. As a consequence, under certain conditions on a finite p-group G \((p>2),\) ( p > 2 ) , the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3,  which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).