 Thompson's coprime character extension theorem

Let $N◃G$ and let $\theta \in \mathrm{Irr}(N)$ be invariant under $G$. If $\theta (1)$ is coprime to $[G:N]$, then $\theta$ extends to $G$ if and only if its determinant character extends to a linear character of $G$. In particular, if the determinant order is also coprime to the index, there is a unique extension whose determinant order is coprime to the index. When $G/N$ is a p-group and $N=O^p(N)$, every invariant irreducible character of degree prime to $p$ satisfies this latter determinant condition.