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<p>In this short note we prove that if <i>G</i> is a perfect locally finite barely transitive group, then the finitary residual of <i>G</i>, namely <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {F}(G)=\{g\in G|~|\text{ supp }(g)|&lt;\omega \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo stretchy="false">|</mo> <mspace width="3.33333pt" /> <mo stretchy="false">|</mo> <mspace width="0.333333em" /> <mtext>supp</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo>&lt;</mo> <mi>ω</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, is trivial. In particular, we prove that there do not exist any perfect locally finite minimal non-<i>FC</i> and minimal non-<i>CC</i>-groups. This completes the description of minimal non-<i>FC</i>-groups and locally graded minimal non-<i>CC</i>-groups. It is a long-standing problem (see [<CitationRef CitationID="CR14">14</CitationRef>, Problem 5.1(b)]).</p>

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Nonexistence of perfect locally finite minimal non-FC and CC-groups, and a characterization of perfect locally finite barely transitive groups

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  • 发表日期：

    2026年1月12日

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  • Ahmet Arikan

摘要

In this short note we prove that if G is a perfect locally finite barely transitive group, then the finitary residual of G, namely \(\mathcal {F}(G)=\{g\in G|~|\text{ supp }(g)|<\omega \}\) F ( G ) = { g ∈ G | | supp ( g ) | < ω } , is trivial. In particular, we prove that there do not exist any perfect locally finite minimal non-FC and minimal non-CC-groups. This completes the description of minimal non-FC-groups and locally graded minimal non-CC-groups. It is a long-standing problem (see [14, Problem 5.1(b)]).