Let \(n=p^ am\), p prime and \(p\nmid m\). Define \(\omega_ p(n)=a\). Let G be a finite group. Define \(\omega_ p(G)=\max \{\omega_ p(\chi (1))|\) \(\chi\in Irr(G)\}\). Let \(l_ p(G)\) be the p-length of G. Extending earlier results of B. Huppert [Math. Z. 67, 479-518 (1957; Zbl 0079.037)] and using theorems of I. M. Isaacs [Pac. J. Math. 36, 677-691 (1971; Zbl 0231.20003)] it is proved that \(l_ p(G)\leq a+\log_ b(c.\omega_ p(G).d+e)\), where 1. \(p\neq 2\) is not a Fermat prime, \(a=3\), \(b=p\), \(d=1\), \(e=0\) or 2. \(p\neq 3\) is a Fermat prime, \(a=2\), \(b=p-2\), \(d=p-3\), \(e=1\) or 3. \(p=3\), \(a=3\), \(b=2\), \(d=1\), \(e=0\) and where \(c=1\) if \(| G|\) is odd, \(c=2\) if G is solvable, \(c=4\) if G is p-solvable.