Computing character degrees via a Galois connection
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2015-03-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://www.theoryofgroups.ir/pdf_6212_edb9e19829eb4a1d2264f3c3f26089ed.html |
| Summary: | 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 × | . |
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| ISSN: | 2251-7650 2251-7669 |