<p>Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-group admitting a splitting automorphism of prime order is locally nilpotent if <Equation ID="Equ1"> <EquationSource Format="TEX">$ \langle g, g^\varphi , \dots , g^{\varphi ^{p-1}} \rangle $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">⟨</mo> <mi>g</mi> <mo>,</mo> <msup> <mi>g</mi> <mi>φ</mi> </msup> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msup> <mi>g</mi> <msup> <mi>φ</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </msup> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </Equation>is nilpotent for every <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g \in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> [<CitationRef CitationID="CR7">7</CitationRef>, Problem 10.59]. We prove that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation> is a periodic residually finite <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-group admitting a splitting automorphism of prime order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation> is nilpotent of class bounded in terms of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p$</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>. This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov’s problem cannot be a Tarski monster.</p>

A residually finite analogue of Kegel’s theorem on splitting automorphisms

期刊论文
出版模式：

开放获取
发表日期：

2026年8月1日

查看全文

Alfonso di Bartolo,
Kıvanç Ersoy,
Giovanni Falcone

摘要

Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a $p'$ $p^{'}$ -group admitting a splitting automorphism of prime order is locally nilpotent if $ \langle g, g^\varphi , \dots , g^{\varphi ^{p-1}} \rangle $ $⟨ g, g^{φ}, \dots, g^{φ^{p - 1}} ⟩$ is nilpotent for every $g \in G$ $g \in G$ [7, Problem 10.59]. We prove that if $G$ $G$ is a periodic residually finite $p'$ $p^{'}$ -group admitting a splitting automorphism of prime order $p,$ $p,$ then $G$ $G$ is nilpotent of class bounded in terms of $p$ $p$ . This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov’s problem cannot be a Tarski monster.