Prime power coverings of groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912162928132096 |
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| author | Giudici, Michael Morgan, Luke Praeger, Cheryl E. |
| author_facet | Giudici, Michael Morgan, Luke Praeger, Cheryl E. |
| contents | For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$, and this conjecture has number theoretic implications for relative Brauer groups of algebraic number fields. We prove the conjecture in the case that the action of $G$ on the set of right cosets of $U$ in $G$ is innately transitive. This includes the case where $U$ is a maximal subgroup of $G$. The proof uses a new bound on the order of a nonabelian finite simple group in terms of its number of classes of elements of prime power order, which in turn depends on the Classification of the Finite Simple Groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15543 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prime power coverings of groups Giudici, Michael Morgan, Luke Praeger, Cheryl E. Group Theory For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$, and this conjecture has number theoretic implications for relative Brauer groups of algebraic number fields. We prove the conjecture in the case that the action of $G$ on the set of right cosets of $U$ in $G$ is innately transitive. This includes the case where $U$ is a maximal subgroup of $G$. The proof uses a new bound on the order of a nonabelian finite simple group in terms of its number of classes of elements of prime power order, which in turn depends on the Classification of the Finite Simple Groups. |
| title | Prime power coverings of groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2412.15543 |