University of IsfahanInternational Journal of Group Theory2251-76504120150301Computing character degrees via a Galois connection16621210.22108/ijgt.2015.6212ENMark L.LewisDepartment of Mathematical Sciences
Kent State UniversityJohn K.McVeyDepartment of Mathematical Sciences Kent State UniversityJournal Article20140609In 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 $C\rtimes{\rm Gal(E/F)}$, 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 $C\leq E^\times$ which satisfies the condition that every prime dividing $|E^\times :C|$ divides $|F^\times|$.https://ijgt.ui.ac.ir/article_6212_edb9e19829eb4a1d2264f3c3f26089ed.pdf