= Thompson's coprime character extension theorem
{c}
Let $N\triangleleft G$ and let $\theta\in\operatorname{Irr}(N)$ 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.
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