A class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_745_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation> of soluble groups is D-bounded when there exists a uniform upper bound for the lengths <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_745_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$d(\varGamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the derived series for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_745_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\varGamma \in \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Γ</mi> <mo>∈</mo> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation>. A theorem of Zassenhaus (Abh. Math. Semin. Hansisch. Univ. 12, 289–312, <CitationRef CitationID="CR13">1938</CitationRef>) states that for each n the class of soluble subgroups of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_745_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$GL(n,\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is D-bounded. Although Zassenhaus’s theorem is fundamental to the study infinite discrete linear groups the proof given here is located within the theory of continuous groups and the only discrete groups which appear are finite.

A Continuous Proof of Zassenhaus’s Solubility Theorem

期刊论文
出版模式：

开放获取
发表日期：

2025年5月14日

查看全文

F. E. A. Johnson

摘要

A class $\mathcal {S}$ $S$ of soluble groups is D-bounded when there exists a uniform upper bound for the lengths $d(\varGamma )$ $d (Γ)$ of the derived series for $\varGamma \in \mathcal {S}$ $Γ \in S$ . A theorem of Zassenhaus (Abh. Math. Semin. Hansisch. Univ. 12, 289–312, 1938) states that for each n the class of soluble subgroups of $GL(n,\mathbb {C})$ $G L (n, C)$ is D-bounded. Although Zassenhaus’s theorem is fundamental to the study infinite discrete linear groups the proof given here is located within the theory of continuous groups and the only discrete groups which appear are finite.