The determination of the minimal generator number d(G) of a polycyclic group G is one of the few still open problems in the algorithmic theory of these groups. If <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2183_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\hat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> is the profinite completion of G, then it is known that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2183_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$d(G) = d(\hat{G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2183_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$d(G) = d(\hat{G}) +1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> holds. The aim here is to introduce a construction for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2183_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$d(\hat{G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="false">^</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if G is polycyclic and nilpotent-by-finite.

On the minimal generator number of a polycyclic nilpotent-by-finite group

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2025年10月7日

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Bettina Eick

摘要

The determination of the minimal generator number d(G) of a polycyclic group G is one of the few still open problems in the algorithmic theory of these groups. If $\hat{G}$ $\hat{G}$ is the profinite completion of G, then it is known that $d(G) = d(\hat{G})$ $d (G) = d (\hat{G})$ or $d(G) = d(\hat{G}) +1$ $d (G) = d (\hat{G}) + 1$ holds. The aim here is to introduce a construction for $d(\hat{G})$ $d (\hat{G})$ if G is polycyclic and nilpotent-by-finite.