Based on the notions of conciseness and semiconciseness, we show that these properties are not equivalent by proving that a word originally presented by Ol’shanskii is semiconcise but not concise. We further establish that every <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2172_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$1/m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>-concise word is semiconcise by proving that when the group-word w takes finitely many values in G, the iterated commutator subgroup <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2172_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$[w(G), G, {\mathop {\dots }\limits ^{(m)}}, G]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mover> <mo>⋯</mo> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mover> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is finite for some <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2172_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$m \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2172_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$[w(G), G]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is finite.

On generalisations of conciseness

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2025年9月9日

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Andoni Zozaya

摘要

Based on the notions of conciseness and semiconciseness, we show that these properties are not equivalent by proving that a word originally presented by Ol’shanskii is semiconcise but not concise. We further establish that every $1/m$ $1 / m$ -concise word is semiconcise by proving that when the group-word w takes finitely many values in G, the iterated commutator subgroup $[w(G), G, {\mathop {\dots }\limits ^{(m)}}, G]$ $[w (G), G, \overset{(m)}{\dots}, G]$ is finite for some $m \in \mathbb {N}$ $m \in N$ if and only if $[w(G), G]$ $[w (G), G]$ is finite.