Finite $p$-groups of class $2$ as central extensions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866911333613568000 |
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| author | Chen, Haimiao |
| author_facet | Chen, Haimiao |
| contents | Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions.
Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and describe the actions of ${\rm End}(G)$ and ${\rm End}(A)$ on $H^2(G,A)$. These are used to provide an efficient criterion for lifting endomorphisms of $G$ to homomorphisms between two central extensions. Subsequently, we present two applications to illustrate the usefulness of this approach, in the case $p>2$.
First, we recover the classification of two-generator $p$-groups of class $2$ up to isomorphism, and compute the order of the automorphism group for each isomorphism class. Second, we construct a family of nonabelian $p$-groups of order $p^7$ whose automorphism groups are abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_10970 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Finite $p$-groups of class $2$ as central extensions Chen, Haimiao Group Theory 20D15, 20D45, 20J05 Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and describe the actions of ${\rm End}(G)$ and ${\rm End}(A)$ on $H^2(G,A)$. These are used to provide an efficient criterion for lifting endomorphisms of $G$ to homomorphisms between two central extensions. Subsequently, we present two applications to illustrate the usefulness of this approach, in the case $p>2$. First, we recover the classification of two-generator $p$-groups of class $2$ up to isomorphism, and compute the order of the automorphism group for each isomorphism class. Second, we construct a family of nonabelian $p$-groups of order $p^7$ whose automorphism groups are abelian. |
| title | Finite $p$-groups of class $2$ as central extensions |
| topic | Group Theory 20D15, 20D45, 20J05 |
| url | https://arxiv.org/abs/2008.10970 |