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arXiv · 2102.06784

Sylow branching coefficients and a conjecture of Malle and Navarro

Eugenio Giannelli·Stacey Law·Jason Long·Carolina Vallejo
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Abstract

We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({\bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.

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Eugenio Giannelli, Stacey Law, Jason Long, Carolina Vallejo. 2021-02-12. Sylow branching coefficients and a conjecture of Malle and Navarro. https://arxiv.org/abs/2102.06784

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