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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2409.03399 |
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| _version_ | 1866910617844056064 |
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| author | Deloup, Florian L. |
| author_facet | Deloup, Florian L. |
| contents | It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg group ${\mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Heisenberg groups Deloup, Florian L. Group Theory 20J06, 20F18 It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G/[G,G]$ to a Heisenberg group ${\mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative. |
| title | On Heisenberg groups |
| topic | Group Theory 20J06, 20F18 |
| url | https://arxiv.org/abs/2409.03399 |