= Isaacs' character-degree criterion for p-nilpotency
{c}
{title2=$\left(\sum_{p\nmid\chi(1)}\chi(1)^2\right)_p=|G:G'|_p$}
The integer $h=|G:G'|_p$ divides $s=\sum_{\chi\in\operatorname{Irr}_{p'}(G)}\chi(1)^2$. The group is <p-nilpotent> exactly when $p\nmid s/h$. To see this, let $N=O^p(G)$. Invariant p-prime-degree characters of $N$ extend, and each has exactly $h$ extensions of p-prime degree. Conjugation by the p-group $G/N$ then gives $s/h\equiv|N|\pmod p$. Thus the quotient is prime to $p$ precisely when $N$ is a <normal p-complement>.
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