On Pro-$p$ Cappitt Groups with finite exponent
A pro-$p$ Cappitt group is a pro-$p$ group $G$ such that $\tilde{S}(G) = \overline{ \langle L \leqslant _c G \:|\, L \ntriangleleft G \rangle }$ is a proper subgroup (i.e. $\tilde{S}(G) \ne G$). In this paper we prove that non-abelian pro-$p$ Cappitt groups whose torsion subgroup is closed and it ha...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.562/ |
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| author | Porto, Anderson Lima, Igor |
| author_facet | Porto, Anderson Lima, Igor |
| author_sort | Porto, Anderson |
| collection | DOAJ |
| description | A pro-$p$ Cappitt group is a pro-$p$ group $G$ such that $\tilde{S}(G) = \overline{ \langle L \leqslant _c G \:|\, L \ntriangleleft G \rangle }$ is a proper subgroup (i.e. $\tilde{S}(G) \ne G$). In this paper we prove that non-abelian pro-$p$ Cappitt groups whose torsion subgroup is closed and it has finite exponent. This result is a natural continuation of main result of the first author [7]. We also prove that in a pro-$p$ Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show that pro-$2$ Cappitt groups of exponent $4$ are pro-$2$ Dedekind groups. These results are pro-$p$ versions of the generalized Dedekind groups studied by Cappitt (see Theorem 1 and Lemma 7 in [1]). |
| format | Article |
| id | doaj-art-3431ddb8b6394fe9a5ca811fd98af216 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3431ddb8b6394fe9a5ca811fd98af2162025-02-07T11:19:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G328729210.5802/crmath.56210.5802/crmath.562On Pro-$p$ Cappitt Groups with finite exponentPorto, Anderson0https://orcid.org/0000-0002-8800-0827Lima, Igor1Departamento de Matemática, Universidade de Brasília, Brasilia-DF, 70910-900 Brazil; Instituto de Ciência e Tecnologia - ICT, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Diamantina - MG, 39100-000 BrazilDepartamento de Matemática, Universidade de Brasília, Brasilia-DF, 70910-900 Brazil; Instituto de Ciência e Tecnologia - ICT, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Diamantina - MG, 39100-000 BrazilA pro-$p$ Cappitt group is a pro-$p$ group $G$ such that $\tilde{S}(G) = \overline{ \langle L \leqslant _c G \:|\, L \ntriangleleft G \rangle }$ is a proper subgroup (i.e. $\tilde{S}(G) \ne G$). In this paper we prove that non-abelian pro-$p$ Cappitt groups whose torsion subgroup is closed and it has finite exponent. This result is a natural continuation of main result of the first author [7]. We also prove that in a pro-$p$ Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show that pro-$2$ Cappitt groups of exponent $4$ are pro-$2$ Dedekind groups. These results are pro-$p$ versions of the generalized Dedekind groups studied by Cappitt (see Theorem 1 and Lemma 7 in [1]).https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.562/Generalized Dedekind groupspro-$p$ Cappitt groupstorsion groups. |
| spellingShingle | Porto, Anderson Lima, Igor On Pro-$p$ Cappitt Groups with finite exponent Comptes Rendus. Mathématique Generalized Dedekind groups pro-$p$ Cappitt groups torsion groups. |
| title | On Pro-$p$ Cappitt Groups with finite exponent |
| title_full | On Pro-$p$ Cappitt Groups with finite exponent |
| title_fullStr | On Pro-$p$ Cappitt Groups with finite exponent |
| title_full_unstemmed | On Pro-$p$ Cappitt Groups with finite exponent |
| title_short | On Pro-$p$ Cappitt Groups with finite exponent |
| title_sort | on pro p cappitt groups with finite exponent |
| topic | Generalized Dedekind groups pro-$p$ Cappitt groups torsion groups. |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.562/ |
| work_keys_str_mv | AT portoanderson onpropcappittgroupswithfiniteexponent AT limaigor onpropcappittgroupswithfiniteexponent |