Sylow branching coefficients and a conjecture of Malle and Navarro

TL;DR

This paper proves a characterization of when a finite group has a normal Sylow p-subgroup based on properties of irreducible characters and Sylow branching coefficients, confirming a conjecture by Malle and Navarro.

Contribution

It establishes a new criterion linking Sylow p-subgroups to character theory, confirming a conjecture by Malle and Navarro from 2012.

Findings

Characterization of normal Sylow p-subgroups via irreducible characters

Reduction to simple groups and combinatorial analysis of Sylow branching coefficients

Confirmation of Malle and Navarro's conjecture from 2012

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.