Computing character degrees via a Galois connection

Document Type: Ischia Group Theory 2014

Authors

1 Department of Mathematical Sciences Kent State University

2 Department of Mathematical Sciences Kent State University

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.
‎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 $|F^\times|$.

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