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

Sylow branching coefficients and a conjecture of Malle and Navarro