On principal indecomposable degrees and Sylow subgroups

Abstract

We conjectured in Malle and Navarro (J Algebra 370:402–406, 2012) that a Sylow p-subgroup P of a finite group G is normal if and only if whenever p does not divide the multiplicity of $$\chi \in {{\text {Irr}}}(G)$$ in the permutation character $$(1_P)^G$$ , then p does not divide the degree $$\chi (1)$$ . In this note, we prove an analogue of this for p-Brauer characters.