Computing character degrees via a Galois connection

Document Type : Ischia Group Theory 2014


Department of Mathematical Sciences Kent State University


‎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 $|F^\times|$.


Main Subjects

I. M. Isaacs (1976). Character Theory of Finite Groups. ‎Academic Press‎, ‎San Diego. O. Manz and T. R. Wolf (1993). Representations of Solvable Groups. ‎Cambridge University Press‎, ‎Cambridge. J. K. McVey (2004). ‎Prime divisibility among degrees of solvable groups. Comm. Algebra. 32, 3391-3402 J. K. McVey (2013). ‎On a Galois connection between the subfield lattice and the multiplicative subgroup lattice. Pacific J‎. ‎Math.. 264, 213-219 J. Riedl (1999). ‎Character degrees‎, ‎class sizes‎, ‎and normal subgroups of a certain class of $p$-groups,. J. Algebra. 218, 190-215 K. Zsigmondy (1892). ‎Zur Theorie der Potenzreste. Monatsh. f. Math.. 3, 265-284
Volume 4, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2014-Part I
March 2015
Pages 1-6
  • Receive Date: 09 June 2014
  • Revise Date: 04 September 2014
  • Accept Date: 03 September 2014
  • Published Online: 01 March 2015