Sylow branching coefficients and a conjecture of Malle and Navarro

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