Given a profinite group G and a family <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> of finite groups closed under taking subgroups, direct products and quotients, denote by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {F}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the set of elements <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g \in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{x \in G\ |\ \langle g,x \rangle \ \text{ is } \text{ a } \text{ pro- }\mathcal {F} \text{ group }\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>G</mi> <mspace width="4pt" /> <mo stretchy="false">|</mo> <mspace width="4pt" /> <mo stretchy="false">⟨</mo> <mi>g</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">⟩</mo> <mspace width="4pt" /> <mspace width="0.333333em" /> <mtext>is</mtext> <mspace width="0.333333em" /> <mspace width="0.333333em" /> <mtext>a</mtext> <mspace width="0.333333em" /> <mspace width="0.333333em" /> <mtext>pro-</mtext> <mspace width="0.333333em" /> <mi mathvariant="script">F</mi> <mspace width="0.333333em" /> <mtext>group</mtext> <mspace width="0.333333em" /> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> has positive Haar measure. We investigate the properties of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {F}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for various choices of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> and the influence of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {F}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on the structure of G when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu (\mathcal F(G))>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Profinite groups with many elements with large nilpotentizer and generalizations

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2025年9月12日

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Martino Garonzi,
Andrea Lucchini,
Nowras Otmen

摘要

Given a profinite group G and a family $\mathcal {F}$ $F$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal {F}(G)$ $F (G)$ the set of elements $g \in G$ $g \in G$ such that $\{x \in G\ |\ \langle g,x \rangle \ \text{ is } \text{ a } \text{ pro- }\mathcal {F} \text{ group }\}$ ${x \in G | ⟨ g, x ⟩ is a pro- F group}$ has positive Haar measure. We investigate the properties of $\mathcal {F}(G)$ $F (G)$ for various choices of $\mathcal {F}$ $F$ and the influence of $\mathcal {F}(G)$ $F (G)$ on the structure of G when $\mu (\mathcal F(G))>0$ $μ (F (G)) > 0$ .