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A normal version of Brauer's height zero conjecture

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arxiv 2406.06428 v2 pith:OJFYCHDT submitted 2024-06-10 math.GR

Alexander Moretó , A. A. Schaeffer Fry This is my paper · ORCID
classification math.GR
keywords conjecturebrauerheightnormalzeroabeliano-michlerpart
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abstract

The celebrated It\^o-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 It\^o-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.

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  1. Character degrees and local subgroups revisited

    math.GR 2024-11 accept novelty 6.0 of 10

    In finite q-solvable groups, the p′-degree irreducible characters are all q′-degree exactly when some Sylow p-subgroup lies inside the normalizer of a Sylow q-subgroup and the derived subgroup of that Sylow q-subgroup...

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