Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _k=[x_1,\dots ,x_k]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the k-th lower central group-word. Given a group G, we write <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X_k(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for the set of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>-values and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma _k(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for the k-th term of the lower central of G. This paper deals with groups in which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\langle g^{X_k(G)} \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>g</mi> <mrow> <msub> <mi>X</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> is a Chernikov group of size at most (m, n) for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$g\in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>. The main result is that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma _{k+1}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a Chernikov group and its size is (k, m, n)-bounded.

On Chernikov-by-nilpotent groups

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2026年3月19日

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Martina Capasso,
Liliana Lancellotti,
Pavel Shumyatsky

摘要

Let $\gamma _k=[x_1,\dots ,x_k]$ $γ_{k} = [x_{1}, \dots, x_{k}]$ be the k-th lower central group-word. Given a group G, we write $X_k(G)$ $X_{k} (G)$ for the set of $\gamma _k$ $γ_{k}$ -values and $\gamma _k(G)$ $γ_{k} (G)$ for the k-th term of the lower central of G. This paper deals with groups in which $\langle g^{X_k(G)} \rangle $ $⟨ g^{X_{k} (G)} ⟩$ is a Chernikov group of size at most (m, n) for all $g\in G$ $g \in G$ . The main result is that $\gamma _{k+1}(G)$ $γ_{k + 1} (G)$ is a Chernikov group and its size is (k, m, n)-bounded.