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A computacional verification of Alperin’s weight conjecture for groups of small order and their prime fields
Verificación computacional de la conjetura de pesos de Alperin para grupos de orden pequeño y sus campos primos
Keywords:
Group representation, Alperin’s conjecture, Weight, Software, Computational, 2000 Mathematics Subject Classification. 20C20 (en)Grupo de representaciones, Conjetura de Alperin, Peso, Computacional (es)
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Abstract. Alperin’s Weight Conjecture was originally formulated for algebraically closed fields (see cite [1]). For some families of groups - such as the symmetric groups - it is known to hold for arbitrary fields (see cite [2]), so it is reasonable to ask whether this conjecture holds for arbitrary fields, and in particular, if it holds for finite fields. We wrote computer software in MAGMA (see cite [8]) to test Alperin’s Weight Conjecture for finite fields, and tested this software on groups of small order and the prime fields whose characteristics divide the order of the groups. We found no counterexamples to this version of Alperin’s Conjecture for groups of order up to 100.
La conjetura de pesos de Alperin fue formulada originalmente para campos algebraicamente cerrados. Para algunas familias de grupos –como por ejemplo los grupos simétricos- esta Conjetura es válida para todos los campos, y en particular, para los campos finitos. E s razonable preguntar si dicha Conjetura permanecerá válida para todos los grupos y todos los campos, y en particular para los campos finitos. En este artículo verificamos (usando MAGMA) la Conjetura de Pesos de Alperin para todos los grupos de orden menor o igual a 100 y todos los campos primos cuyas características dividen el orden de cada grupo.
References
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