Comptes Rendus
Research article - Group theory
On the factorised subgroups of products of cyclic and non-cyclic finite p-groups
Brendan McCann1
1 Department of Computing and Mathematics, South-East Technological University Waterford, Cork Road, Waterford, Ireland
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 293-300.

Let p be an odd prime and let G=AB be a finite p-group that is the product of a cyclic subgroup A and a non-cyclic subgroup B. Suppose in addition that the nilpotency class of B is less than p 2. We denote by ℧ i (B) the subgroup of B generated by the p i -th powers of elements of B, that is ℧ i (B)=〈b p i ∣b∈B〉. In this article we show that, for all values of i, the set A℧ i (B) is a subgroup of G. We also present some applications of this result.

Soient p un nombre premier impair et G=AB un p-groupe fini qui est le produit des sous-groupes A et B, tels que A soit un sous-groupe cyclique et B soit un sous-groupe non cyclique. Supposons également que la classe de nilpotence de B soit inférieure à p 2. On note ℧ i (B) le sous-groupe de B engendré par les puissances p i des éléments de B, alors ℧ i (B)=〈b p i ∣b∈B〉. Dans cet article nous montrons que, pour chaque valeur du nombre i, l’ensemble A℧ i (B) est sous-groupe du groupe G. Nous présentons également quelques applications de ce résultat.

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Published online:
DOI: 10.5802/crmath.565
Classification: 20D40, 20D15
Keywords: factorised groups, products of groups, finite $p$-groups
Mots-clés : groupes factorisés, produit de groupes, $p$-groupes finis

Brendan McCann 1

1 Department of Computing and Mathematics, South-East Technological University Waterford, Cork Road, Waterford, Ireland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Brendan McCann. On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 293-300. doi : 10.5802/crmath.565. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.565/

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