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Computing character degrees via a Galois connection | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 4، شماره 1، خرداد 2015، صفحه 1-6 اصل مقاله (188.48 K) | ||
| نوع مقاله: Ischia Group Theory 2014 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2015.6212 | ||
| نویسندگان | ||
| Mark L. Lewis* 1؛ John K. McVey2 | ||
| 1Department of Mathematical Sciences Kent State University | ||
| 2Department of Mathematical Sciences Kent State University | ||
| چکیده | ||
| 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 $|F^\times|$. | ||
| کلیدواژه‌ها | ||
| Galois correspondence؛ lattice؛ character degree؛ finite field | ||
| مراجع | ||
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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
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