On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups
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 $\frac{p}{2}$. We denote by $\mho _i(B) $ the subgroup of $B$ generated by the $p^i$-th power...
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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.565/ |
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| author | McCann, Brendan |
| author_facet | McCann, Brendan |
| author_sort | McCann, Brendan |
| collection | DOAJ |
| description | 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 $\frac{p}{2}$. We denote by $\mho _i(B) $ the subgroup of $B$ generated by the $p^i$-th powers of elements of $B$, that is $ \mho _i(B) = \langle b^{p^i} \mid b \in B \rangle $. In this article we show that, for all values of $i$, the set $ A \mho _i(B) $ is a subgroup of $G$. We also present some applications of this result. |
| format | Article |
| id | doaj-art-4e8ccdbd1bd4410cb9500e6df4446f10 |
| 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-4e8ccdbd1bd4410cb9500e6df4446f102025-02-07T11:19:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G329330010.5802/crmath.56510.5802/crmath.565On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groupsMcCann, Brendan0Department of Computing and Mathematics, South-East Technological University Waterford, Cork Road, Waterford, IrelandLet $p$ 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 $\frac{p}{2}$. We denote by $\mho _i(B) $ the subgroup of $B$ generated by the $p^i$-th powers of elements of $B$, that is $ \mho _i(B) = \langle b^{p^i} \mid b \in B \rangle $. In this article we show that, for all values of $i$, the set $ A \mho _i(B) $ is a subgroup of $G$. We also present some applications of this result.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.565/factorised groupsproducts of groupsfinite $p$-groups |
| spellingShingle | McCann, Brendan On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups Comptes Rendus. Mathématique factorised groups products of groups finite $p$-groups |
| title | On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups |
| title_full | On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups |
| title_fullStr | On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups |
| title_full_unstemmed | On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups |
| title_short | On the factorised subgroups of products of cyclic and non-cyclic finite $p$-groups |
| title_sort | on the factorised subgroups of products of cyclic and non cyclic finite p groups |
| topic | factorised groups products of groups finite $p$-groups |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.565/ |
| work_keys_str_mv | AT mccannbrendan onthefactorisedsubgroupsofproductsofcyclicandnoncyclicfinitepgroups |