Virtually nilpotent groups with finitely many orbits under automorphisms
| Autor: | Alex C. Dantas, Emerson de Melo, Raimundo Bastos |
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| Rok vydání: | 2020 |
| Předmět: |
Torsion subgroup
Group (mathematics) General Mathematics 010102 general mathematics Group Theory (math.GR) Automorphism 01 natural sciences Omega Combinatorics Nilpotent Mathematics::Group Theory Mathematics::K-Theory and Homology 0103 physical sciences 20E22 20E36 FOS: Mathematics 010307 mathematical physics 0101 mathematics Characteristic subgroup Nilpotent group Mathematics - Group Theory Mathematics |
| DOI: | 10.48550/arxiv.2008.10800 |
| Popis: | Let G be a group. The orbits of the natural action of $${{\,\mathrm{Aut}\,}}(G)$$ on G are called automorphism orbits of G, and the number of automorphism orbits of G is denoted by $$\omega (G)$$ . Let G be a virtually nilpotent group such that $$\omega (G)< \infty $$ . We prove that $$G = K \rtimes H$$ where H is a torsion subgroup and K is a torsion-free nilpotent radicable characteristic subgroup of G. Moreover, we prove that $$G^{'}= D \times {{\,\mathrm{Tor}\,}}(G^{'})$$ where D is a torsion-free nilpotent radicable characteristic subgroup. In particular, if the maximum normal torsion subgroup $$\tau (G)$$ of G is trivial, then $$G^{'}$$ is nilpotent. |
| Databáze: | OpenAIRE |
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