A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911894811443200 |
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| author | Murashka, V. I. |
| author_facet | Murashka, V. I. |
| contents | L.A. Shemetkov posed a Problem 9.74 in Kourovka Notebook to find all local formations $\mathfrak{F}$ of finite groups such that every finite minimal non-$\mathfrak{F}$-group is either a Schmidt group or a group of prime order. All known solutions to this problem are obtained under the assumption that every minimal non-$\mathfrak{F}$-group is soluble. Using the above mentioned solutions we present a polynomial in $n$ time check for a local formation $\mathfrak{F}$ with bounded $π(\mathfrak{F})$ to be a formation of soluble groups with the Shemtkov property where $n=\max π(\mathfrak{F})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property Murashka, V. I. Group Theory 20D10, 20F19 L.A. Shemetkov posed a Problem 9.74 in Kourovka Notebook to find all local formations $\mathfrak{F}$ of finite groups such that every finite minimal non-$\mathfrak{F}$-group is either a Schmidt group or a group of prime order. All known solutions to this problem are obtained under the assumption that every minimal non-$\mathfrak{F}$-group is soluble. Using the above mentioned solutions we present a polynomial in $n$ time check for a local formation $\mathfrak{F}$ with bounded $π(\mathfrak{F})$ to be a formation of soluble groups with the Shemtkov property where $n=\max π(\mathfrak{F})$. |
| title | A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property |
| topic | Group Theory 20D10, 20F19 |
| url | https://arxiv.org/abs/2405.20257 |