On 𝑝-parts of character codegrees

Abstract

Abstract Let 𝐺 be a finite group and 𝑝 a prime. We define the codegree of χ ∈ Irr ⁢ ( G ) \chi\in\mathrm{Irr}(G) by cod ( χ ) = | G : ker χ | / χ ( 1 ) \operatorname{cod}(\chi)=\lvert G:\ker\chi\rvert/\chi(1) and define c p ( G ) = max { log p ( cod ( χ ) ) p ∣ χ ∈ Irr ( G ) } c_{p}(G)=\max\{\log_{p}(\operatorname{cod}(\chi))_{p}\mid\chi\in\mathrm{Irr}(G)\} . In this paper, we show that | G / O p ′ ⁢ p ⁢ ( G ) | p ≤ p c p ⁢ ( G ) \lvert G/O_{p^{\prime}p}(G)\rvert_{p}\leq p^{c_{p}(G)} for all finite groups 𝐺 and characterize the finite groups 𝐺 with c p ⁢ ( G ) ≤ 1 c_{p}(G)\leq 1 .