= Gluing determinant-normalized character extensions
{title2=$f(g)=\chi_{N\langle g\rangle}(g)$}
Under the invariant, coprime-degree and determinant hypotheses of the <coprime-degree determinant extension theorem>, without assuming $G/N$ solvable, each cyclic-quotient subgroup $N\langle g\rangle$ has a unique determinant-normalized extension. Use its value at $g$ to define $f$. Uniqueness makes $f$ conjugacy-invariant. On an <elementary subgroup> $E$, it agrees with the restriction of the extension to $NE$, since $NE/N$ is a solvable quotient of $E$. <Brauer's characterization of characters> therefore makes $f$ a <generalized character>, with $f_N=\theta$.
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