<p>We study the Wamsley group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_949_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="320" /> </InlineMediaObject> <EquationSource Format="TEX">$\langle X,Y,Z\,|\, X^Z=X^\alpha , ^Z Y=Y^\beta , Z^\gamma =[X,Y]\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <msup> <mi>X</mi> <mi>Z</mi> </msup> <mo>=</mo> <msup> <mi>X</mi> <mi>α</mi> </msup> <msup> <mo>,</mo> <mi>Z</mi> </msup> <mi>Y</mi> <mo>=</mo> <msup> <mi>Y</mi> <mi>β</mi> </msup> <mo>,</mo> <msup> <mi>Z</mi> <mi>γ</mi> </msup> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> and its Sylow subgroups, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_949_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha ^\gamma \ne 1\ne \beta ^\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>α</mi> <mi>γ</mi> </msup> <mo>≠</mo> <mn>1</mn> <mo>≠</mo> <msup> <mi>β</mi> <mi>γ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_949_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, obtaining the sharpest results when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_949_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

A Study of the Wamsley Group and Its Sylow Subgroups

期刊论文
发表日期：

2025年1月22日

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Andrea Previtali,
Fernando Szechtman

摘要

We study the Wamsley group $\langle X,Y,Z\,|\, X^Z=X^\alpha , ^Z Y=Y^\beta , Z^\gamma =[X,Y]\rangle $ $⟨ X, Y, Z | X^{Z} = X^{α},^{Z} Y = Y^{β}, Z^{γ} = [X, Y] ⟩$ and its Sylow subgroups, where $\alpha ^\gamma \ne 1\ne \beta ^\gamma $ $α^{γ} \neq 1 \neq β^{γ}$ and $\gamma >0$ $γ > 0$ , obtaining the sharpest results when $\alpha =\beta $ $α = β$ .