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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.01011 |
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| _version_ | 1866915906020442112 |
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| author | Di Bartolo, Alfonso Ersoy, Kıvanç Falcone, Giovanni |
| author_facet | Di Bartolo, Alfonso Ersoy, Kıvanç Falcone, Giovanni |
| contents | 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 \(p'\)-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^φ, \dots, g^{φ^{p-1}} \rangle \] is nilpotent for every \(g \in G\), \cite[Problem 10.59]{kourovka21}. We prove that if \(G\) is a periodic residually finite group admitting a splitting automorphism of prime order \(p\) then \(G\) is nilpotent of class bounded in terms of \(p\). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01011 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A residually finite analogue of Kegel's theorem on splitting automorphisms Di Bartolo, Alfonso Ersoy, Kıvanç Falcone, Giovanni Group Theory 20E36 20F50 20F28 20D45 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 \(p'\)-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^φ, \dots, g^{φ^{p-1}} \rangle \] is nilpotent for every \(g \in G\), \cite[Problem 10.59]{kourovka21}. We prove that if \(G\) is a periodic residually finite group admitting a splitting automorphism of prime order \(p\) then \(G\) is nilpotent of class bounded in terms of \(p\). 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. |
| title | A residually finite analogue of Kegel's theorem on splitting automorphisms |
| topic | Group Theory 20E36 20F50 20F28 20D45 |
| url | https://arxiv.org/abs/2604.01011 |