COMPUTING CHARACTER DEGREES VIA A GALOIS CONNECTION

In a previous paper, the second author established that, given finite fields F < E and certain subgroups C <= E-X, there is a Galois connection between the intermediate field lattice {L vertical bar F <= L <= 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 (sic) Gal(E/F). However, the analysis when vertical bar F vertical bar is a Mersenne prime is more complicated, so certain cases were omitted from that paper. 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 C (sic) 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 C < E-X which satisfies the condition that every prime dividing vertical bar E-X : C vertical bar divides vertical bar F-X vertical bar.