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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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.565/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |