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Title

Computing character degrees via a Galois connection

Year: 1394
COI: JR_THEGR-4-1_002
Language: EnglishView: 18
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Authors

Mark Lewis - Department of Mathematical Sciences Kent State University
John McVey - 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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Paper COI Code

This Paper COI Code is JR_THEGR-4-1_002. Also You can use the following address to link to this article. This link is permanent and is used as an article registration confirmation in the Civilica reference:

https://civilica.com/doc/1199979/

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Lewis, Mark and McVey, John,1394,Computing character degrees via a Galois connection,https://civilica.com/doc/1199979

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  • I. M. Isaacs (1976). Character Theory of Finite Groups. ‎Academic ...
  • O. Manz and T. R. Wolf (1993). Representations of Solvable ...
  • J. K. McVey (2004). ‎Prime divisibility among degrees of solvable ...
  • J. K. McVey (2013). ‎On a Galois connection between the ...
  • J. Riedl (1999). ‎Character degrees‎, ‎class sizes‎, ‎and normal subgroups ...
  • K. Zsigmondy (1892). ‎Zur Theorie der Potenzreste. Monatsh. f. Math.. ...
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