Computing character degrees via a Galois connection

Abstract

‎In a previous paper‎, ‎the second author established that‎, ‎given finite fields $F < E$ and certain subgroups $C leq E^times$‎, ‎there is a Galois connection between the intermediate field lattice ${L mid F leq L leq E}$ and $C$'s subgroup lattice‎. ‎Based on the Galois connection‎, ‎the paper then calculated the irreducible‎, ‎complex character degrees of the semi-direct product $C rtimes {Gal} (E/F)$‎. ‎However‎, ‎the analysis when $|F|$ is a Mersenne prime is more complicated‎, ‎so certain cases were omitted from that paper‎. <br />‎The present exposition‎, ‎which is a reworking of the previous article‎, ‎provides a uniform analysis over all the families‎, ‎including the previously undetermined ones‎. ‎In the group $Crtimes{rm Gal(E/F)}$‎, ‎we use the Galois connection to calculate stabilizers of linear characters‎, ‎and these stabilizers determine the full character degree set‎. ‎This is shown for each subgroup $Cleq E^times$ which satisfies the condition that every prime dividing $|E^times‎ :‎C|$ divides $|F^times|$.