Sylow branching coefficients and a conjecture of Malle and Navarro

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 (1P)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.