Minimum $\mathcal{F}$-covers: the supersolvable and metabelian cases
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915779139600384 |
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| author | Lazorec, Mihai-Silviu |
| author_facet | Lazorec, Mihai-Silviu |
| contents | Given a set $\mathcal{F}$ of finite groups, it is said that a group $G$ is an $\mathcal{F}$-cover if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. Moreover, $G$ is a minimum $\mathcal{F}$-cover if there is no $\mathcal{F}$-cover whose order is less than $|G|$. In [Cameron P. J., et al., Minimal cover groups, J. Algebra 660 (2024)], the authors pose the following question: For which classes $\mathcal{X}$ of groups, closed under taking subgroups and direct products, is it true that, if $\mathcal{F}$ is a set of $\mathcal{X}$-groups, then there is a minimum $\mathcal{F}$-cover which is an $\mathcal{X}$-group? In this paper, we give a negative answer in two cases: $\mathcal{X}\in \{``supersolvable", ``metabelian"\}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06220 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimum $\mathcal{F}$-covers: the supersolvable and metabelian cases Lazorec, Mihai-Silviu Group Theory Given a set $\mathcal{F}$ of finite groups, it is said that a group $G$ is an $\mathcal{F}$-cover if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. Moreover, $G$ is a minimum $\mathcal{F}$-cover if there is no $\mathcal{F}$-cover whose order is less than $|G|$. In [Cameron P. J., et al., Minimal cover groups, J. Algebra 660 (2024)], the authors pose the following question: For which classes $\mathcal{X}$ of groups, closed under taking subgroups and direct products, is it true that, if $\mathcal{F}$ is a set of $\mathcal{X}$-groups, then there is a minimum $\mathcal{F}$-cover which is an $\mathcal{X}$-group? In this paper, we give a negative answer in two cases: $\mathcal{X}\in \{``supersolvable", ``metabelian"\}.$ |
| title | Minimum $\mathcal{F}$-covers: the supersolvable and metabelian cases |
| topic | Group Theory |
| url | https://arxiv.org/abs/2602.06220 |