Sylow branching coefficients and a conjecture of Malle and Navarro

Abstract

We prove that a finite group G $G$ has a normal Sylow p $p$ -subgroup P $P$ if, and only if, every irreducible character of G $G$ appearing in the permutation character ( 1 P ) G $({\bf 1}_P)^G$ with multiplicity coprime to p $p$ has degree coprime to p $p$ . 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.