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≤E × , there is a Galois connection between the intermediate field lattice {L∣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⋊Gal(E/F) . However, the analysis when |F| 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⋊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 × which satisfies the condition that every prime dividing |E × :C| divides |F × | .

Keywords

• Galois correspondence
• lattice
• character degree
• finite field