A converse for a theorem of Gallagher

Abstract

Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\ddot{I}‡\in \mathrm{Irr}(G)$ satisfies $\ddot{I}{‡}_N$ is irreducible, then $\ddot{I}‡{\hat{I}}^2$ is irreducible and distinct for all ${\hat{I}}^2\in \mathrm{Irr}(G/N)$. Furthermore, if $\hat{I}¸=\ddot{I}{‡}_N$, then these are all of the irreducible constituents of $\hat{I}{¸}^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\ddot{I}€$-partial characters and that a partial converse of that theorem is true.