On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups

Abstract

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in \mathrm{Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi (1{)}_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.