On P-Parts Of Character Codegrees

Let G be a finite group and p a prime. We define the codegree of chi is an element of Irr(G) by cod(chi) = vertical bar G : ker chi vertical bar/chi(1) and define c(p)(G) = max {log (p)(cod (chi))(p) vertical bar chi is an element of Irr(G)}. In this paper, we show that vertical bar G/O p' p(G)/vertical bar(p) <= p(cp)(G) for all finite groups G and characterize the finite groups G with c(p)(G) <= 1.