Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient
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arXiv
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| Natura: | Preprint |
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2023
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| author | Kosakowska, Justyna Schmidmeier, Markus Schreiner, Martin |
| author_facet | Kosakowska, Justyna Schmidmeier, Markus Schreiner, Martin |
| contents | In his 1934 paper, G.\ Birkhoff poses the problem of classifying pairs $(G,U)$ where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms of $G$. In general, Birkhoff's Problem is not considered feasible. In this note, we fix a prime number $p$ and assume that $G$ is a direct sum of cyclic $p$-groups and $U\subset G$ is a subgroup. Under the assumption that the factor group $G/U$ is an elementary abelian $p$-group, we show that the pair $(G,U)$ always has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$ or $(\mathbb Z/(p^n), (p))$. Surprisingly, in the dual situation we need an additional condition. If we assume that $U$ itself is an elementary subgroup of $G$, then we show that the pair $(G,U)$ has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n), (p^{n-1}))$ if and only if $G/U$ is a~direct sum of cyclic $p$-groups. We generalize the above results to modules over commutative discrete valuation rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01451 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient Kosakowska, Justyna Schmidmeier, Markus Schreiner, Martin Group Theory Representation Theory 20K27, 13C05, 47A15 In his 1934 paper, G.\ Birkhoff poses the problem of classifying pairs $(G,U)$ where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms of $G$. In general, Birkhoff's Problem is not considered feasible. In this note, we fix a prime number $p$ and assume that $G$ is a direct sum of cyclic $p$-groups and $U\subset G$ is a subgroup. Under the assumption that the factor group $G/U$ is an elementary abelian $p$-group, we show that the pair $(G,U)$ always has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$ or $(\mathbb Z/(p^n), (p))$. Surprisingly, in the dual situation we need an additional condition. If we assume that $U$ itself is an elementary subgroup of $G$, then we show that the pair $(G,U)$ has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n), (p^{n-1}))$ if and only if $G/U$ is a~direct sum of cyclic $p$-groups. We generalize the above results to modules over commutative discrete valuation rings. |
| title | Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient |
| topic | Group Theory Representation Theory 20K27, 13C05, 47A15 |
| url | https://arxiv.org/abs/2312.01451 |