 Thompson's character-degree normal complement theorem

If a prime $p$ divides the degree of every nonlinear irreducible character of a finite group, the group has a normal p-complement. Indeed, the p-prime-degree characters are exactly the linear characters. Their squared-degree sum is $|G:G^{\prime }|$, whose p-part is $|G:G^{\prime }{|}_p$; Isaacs' character-degree criterion for p-nilpotency applies.