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Abstract <p> We introduce a group<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm {ECT}(\mathbb {Z})\)</EquationSource> </InlineEquation> generated by generalized class transpositions. We show that the Kohlautomorphism of the group<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm {CT}(\mathbb {Z})\)</EquationSource> </InlineEquation> is induced by an inner automorphism of the group<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm {Sym}(\mathbb {Z})\)</EquationSource> </InlineEquation> and can be represented as the composition of two automorphisms of<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm {Sym}(\mathbb {Z})\)</EquationSource> </InlineEquation>; namely, shift by<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\)</EquationSource> </InlineEquation> and reflection with respect to<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(0\)</EquationSource> </InlineEquation>. We suggest formulas for the action of these automorphisms onthe generators of<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5133_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm {ECT}(\mathbb {Z})\)</EquationSource> </InlineEquation>.</p>

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The Group of Generalized Class Transpositions and its Automorphisms

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • M. V. Neshchadim,
  • S. M. Neshchadim

摘要

Abstract

We introduce a group \(\mathrm {ECT}(\mathbb {Z})\) generated by generalized class transpositions. We show that the Kohlautomorphism of the group \(\mathrm {CT}(\mathbb {Z})\) is induced by an inner automorphism of the group \(\mathrm {Sym}(\mathbb {Z})\) and can be represented as the composition of two automorphisms of \(\mathrm {Sym}(\mathbb {Z})\) ; namely, shift by \(1\) and reflection with respect to \(0\) . We suggest formulas for the action of these automorphisms onthe generators of \(\mathrm {ECT}(\mathbb {Z})\) .