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 |
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| Online Access: | http://www.theoryofgroups.ir/pdf_6212_edb9e19829eb4a1d2264f3c3f26089ed.html |
| _version_ | 1818546493010739200 |
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| author | Mark L. Lewis John K. McVey |
| author_facet | Mark L. Lewis John K. McVey |
| author_sort | Mark L. Lewis |
| collection | DOAJ |
| description | 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 × | . |
| first_indexed | 2024-12-12T07:53:57Z |
| format | Article |
| id | doaj.art-2c935013ffd449e6b00c1904c1ed6fc4 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-12T07:53:57Z |
| publishDate | 2015-03-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-2c935013ffd449e6b00c1904c1ed6fc42022-12-22T00:32:22ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692015-03-014116Computing character degrees via a Galois connectionMark L. Lewis 0John K. McVey 1Department of Mathematical Sciences Kent State UniversityDepartment of Mathematical Sciences Kent State UniversityIn 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 × | .http://www.theoryofgroups.ir/pdf_6212_edb9e19829eb4a1d2264f3c3f26089ed.htmlGalois correspondencelatticecharacter degreefinite field |
| spellingShingle | Mark L. Lewis John K. McVey Computing character degrees via a Galois connection International Journal of Group Theory Galois correspondence lattice character degree finite field |
| title | Computing character degrees via a Galois connection |
| title_full | Computing character degrees via a Galois connection |
| title_fullStr | Computing character degrees via a Galois connection |
| title_full_unstemmed | Computing character degrees via a Galois connection |
| title_short | Computing character degrees via a Galois connection |
| title_sort | computing character degrees via a galois connection |
| topic | Galois correspondence lattice character degree finite field |
| url | http://www.theoryofgroups.ir/pdf_6212_edb9e19829eb4a1d2264f3c3f26089ed.html |
| work_keys_str_mv | AT markllewis computingcharacterdegreesviaagaloisconnection AT johnkmcvey computingcharacterdegreesviaagaloisconnection |