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A lattice theoretic characterization for the existence of a faithful irreducible representation | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 13، شماره 4، اسفند 2024، صفحه 345-350 اصل مقاله (373.72 K) | ||
| نوع مقاله: Ischia Group Theory 2022 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2023.138541.1862 | ||
| نویسنده | ||
| Peter P. Palfy* | ||
| Alfréd Rényi Institute of Mathematics, P.O.Box 127, 1364 Budapest, Hungary | ||
| چکیده | ||
| In a recent article S'ebastien Palcoux formulated a sufficient condition on the subgroup lattice of a finite group $G$ that guarantees the existence of a faithful irreducible complex representation of $G$, and asked whether his condition is also necessary. In this short note we give an affirmative answer using Kochend\"orffer's criterion for the existence of a faithful irreducible representation based on the structure of the socle of $G$. | ||
| کلیدواژهها | ||
| Faithful irreducible representation؛ subgroup lattice؛ Boolean lattice | ||
| مراجع | ||
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[1] W. Burnside, Theory of groups of finite order, Cambridge University Press, 1911. [2] W. Gaschütz, Endliche Gruppen mit treuen absolut-irreduziblen Darstellungen, Math. Nachr., 12 (1954) 253–255. [3] R. Kochendörffer, Über treue irreduzible Darstellungen endlicher Gruppen, Math. Nachr., 1 (1948) 25–39. [4] S. Palcoux, Dual Ore’s theorem on distributive intervals of finite groups, J. Algebra, 505 (2018) 279–287. [5] P. P. Pálfy, On faithful irreducible representations of finite groups, Stud. Sci. Math. Hung., 14 (1979) 95–98. [6] R. Remak, Über minimale invariante Untergruppen in der Theorie der endlichen Gruppen, J. Reine Angew. Math., 162 (1930) 1–16. [7] K. Shoda, Über direkt zerlegbare Gruppen, J. Fac. Sci. Imp. Univ. Tokyo, 2 (1930) 51–72. [8] L. Weisner, Condition that a finite group be multiply isomorphic with each of its irreducible representations, Amer. J. Math., 61 (1939) 709–712. [9] E. M. Zhmud’, On isomorphic linear representations of finite groups (Russian), Mat. Sb. N.S., 38(80) (1956) 417–430. | ||
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آمار تعداد مشاهده مقاله: 98 تعداد دریافت فایل اصل مقاله: 128 |
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