no. G3
Article de recherche - Théorie des groupes
On Pro-p Cappitt Groups with finite exponent
Porto, Anderson1, 2 orcid ; Lima, Igor1, 2 
1Instituto de Ciência e Tecnologia - ICT, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Diamantina - MG, 39100-000 Brazil
2Departamento de Matemática, Universidade de Brasília, Brasilia-DF, 70910-900 Brazil
Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 287-292

A pro-p Cappitt group is a pro-p group G such that S ˜(G)=〈L⩽ c G|L⋪G〉 ¯ is a proper subgroup (i.e. S ˜(G)≠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]).

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.562
Classification : 20E34, 20E18
Keywords: Generalized Dedekind groups, pro-$p$ Cappitt groups, torsion groups.

Porto, Anderson  1 , 2   ; Lima, Igor  1 , 2

1 Instituto de Ciência e Tecnologia - ICT, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Diamantina - MG, 39100-000 Brazil
2 Departamento de Matemática, Universidade de Brasília, Brasilia-DF, 70910-900 Brazil
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2024__362_G3_287_0,
     author = {Porto, Anderson and Lima, Igor},
     title = {On {Pro-}$p$ {Cappitt} {Groups} with finite exponent},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {287--292},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G3},
     doi = {10.5802/crmath.562},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.562/}
}
TY  - JOUR
AU  - Porto, Anderson
AU  - Lima, Igor
TI  - On Pro-$p$ Cappitt Groups with finite exponent
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 287
EP  - 292
VL  - 362
IS  - G3
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.562/
DO  - 10.5802/crmath.562
LA  - en
ID  - CRMATH_2024__362_G3_287_0
ER  - 
%0 Journal Article
%A Porto, Anderson
%A Lima, Igor
%T On Pro-$p$ Cappitt Groups with finite exponent
%J Comptes Rendus. Mathématique
%D 2024
%P 287-292
%V 362
%N G3
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.562/
%R 10.5802/crmath.562
%G en
%F CRMATH_2024__362_G3_287_0
Porto, Anderson; Lima, Igor. On Pro-$p$ Cappitt Groups with finite exponent. Comptes Rendus. Mathématique, Tome 362 (2024) no. G3, pp. 287-292. doi: 10.5802/crmath.562

[1] Cappitt, D. Generalized Dedekind groups, J. Algebra, Volume 17 (1971), pp. 310-316 | Zbl | DOI | MR

[2] The GAP Group GAP — Groups, Algorithms, and Programming, Version 4.10.2, 2019 (http://www.gap-system.org)

[3] Herfort, Wolfgang N. Compact torsion groups and finite exponent, Arch. Math., Volume 33 (1980), pp. 404-410 | DOI | MR

[4] Herfort, Wolfgang N. An arithmetic property of profinite groups, Manuscr. Math., Volume 37 (1982), pp. 11-17 | DOI | MR

[5] Kappe, Luise-Charlotte; Reboli, Denise M. On the structure of generalized Hamiltonian groups, Arch. Math., Volume 75 (2000) no. 5, pp. 328-337 | DOI | MR

[6] Kargapolov, Mikhail I.; Merzljakov, Ju. I. Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, 62, Springer, 1979 | Zbl | DOI

[7] Porto, Anderson L. P. Profinite Cappitt groups, Quaest. Math., Volume 44 (2021) no. 3, pp. 307-311 | DOI | MR | Zbl

[8] Porto, Anderson L. P.; de Bessa, Vagner R. Profinite Dedekind groups, Far East J. Math. Sci., Volume 126 (2020) no. 1, pp. 89-97

[9] Ribes, Luis; Zalesskii, Pavel Profinite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 40, Springer, 2010 | DOI

[10] Robinson, Derek J. S. A course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer, 1996 | DOI

[11] Rotman, Joseph J. An introduction to the theory of Groups, Graduate Texts in Mathematics, 148, Springer, 1995 | Zbl | DOI

[12] Wilson, John S. Profinite Groups, London Mathematical Society Monographs. New Series, Clarendon Press, 1998 | DOI

[13] Zelʼmanov, Efim I. On periodic compact groups, Isr. J. Math., Volume 77 (1992) no. 1-2, pp. 83-95 | DOI | MR

Cité par Sources :