On the width of $\sigma $-subnormality in finite groups

Adolfo Ballester-Bolinches; Sergey Kamornikov; Vicent Pérez-Calabuig; Xiaolan Yi

doi:10.5802/crmath.772

Research article - Algebra

On the width of $\sigma $-subnormality in finite groups

Adolfo Ballester-Bolinches ¹; Sergey Kamornikov ²; Vicent Pérez-Calabuig ¹; Xiaolan Yi ³

¹Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain
²Francisk Skorina State Gomel University, 104 Sovetskaya Str., 246019, Gomel, Belarus
³Zhejiang Sci-Tech University, 310018, Hangzhou, P. R. China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 861-866

Abstract (Original language)
Résumé

Let $\sigma = \bigl \lbrace \sigma _i \;\big \vert \; i \in I\bigr \rbrace $ be a partition of the set of all primes with $\sigma \ne \lbrace \mathbb{P}\rbrace $. Problem 19.84 of the Kourovka Notebook shows that a $\sigma $-analogue of the classical Wielandt criterion of subnormality is not true in general. In this paper, we prove a Wielandt-type $\sigma $-subnormality criterion in terms of the Baer–Suzuki width of the class of all $\sigma _0$ groups, where $\sigma _0$ is the member of $\sigma $ such that $2 \in \sigma _0$.

Soit $\sigma = \bigl \lbrace \sigma _i \;\big \vert \; i \in I\bigr \rbrace $ une partition de l’ensemble des nombres premiers de $\sigma \ne \lbrace \mathbb{P}\rbrace $. Le problème 19.84 du cahier Kourovka montre qu’un $\sigma $-analogue du critère de sous-normalité classique de Wielandt n’est pas vrai en général. Dans cet article, nous prouvons un critère de sous-normalité de type Wielandt en termes de largeur de Baer–Suzuki de la classe de tous les groupes $\sigma _0$, où $\sigma _0$ est le membre de $\sigma $ tel que $2 \in \sigma _0$.

Received: 2025-04-25
Accepted: 2025-06-20
Published online: 2025-07-15

DOI: 10.5802/crmath.772

Classification: 20D10, 20D20, 20D35
Keywords: Finite group, $\sigma $-subnormal subgroup, subnormal subgroup, $\sigma $-nilpotent group, Baer–Suzuki theorem
Mots-clés : Groupe fini, sous-groupe $\sigma $-sous-normal, sous-groupe sous-normal, groupe $\sigma $-nilpotent, théorème de Baer–Suzuki

Author's affiliations:

Adolfo Ballester-Bolinches ¹ ; Sergey Kamornikov ² ; Vicent Pérez-Calabuig ¹ ; Xiaolan Yi ³

¹ Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain
² Francisk Skorina State Gomel University, 104 Sovetskaya Str., 246019, Gomel, Belarus
³ Zhejiang Sci-Tech University, 310018, Hangzhou, P. R. China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Adolfo Ballester-Bolinches and Sergey Kamornikov and Vicent P\'erez-Calabuig and Xiaolan Yi},
     title = {On the width of $\sigma $-subnormality in finite groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {861--866},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.772},
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Adolfo Ballester-Bolinches; Sergey Kamornikov; Vicent Pérez-Calabuig; Xiaolan Yi. On the width of $\sigma $-subnormality in finite groups. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 861-866. doi: 10.5802/crmath.772

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