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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.06428 |
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| _version_ | 1866909224506753024 |
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| author | Moretó, Alexander Fry, A. A. Schaeffer |
| author_facet | Moretó, Alexander Fry, A. A. Schaeffer |
| contents | The celebrated Itô-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the Itô-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A normal version of Brauer's height zero conjecture Moretó, Alexander Fry, A. A. Schaeffer Group Theory The celebrated Itô-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the Itô-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture. |
| title | A normal version of Brauer's height zero conjecture |
| topic | Group Theory |
| url | https://arxiv.org/abs/2406.06428 |