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15: Los teoremas de Sylow

Última actualización

30 oct 2022
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- 14.9: Ejercicios de salvia
- 15.1: Los teoremas de Sylow

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Ya sabemos que lo contrario del Teorema de Lagrange es falso. Si $G$ es un grupo de orden $m$ y $n$ divide $m\text{,}$ entonces $G$ no necesariamente posee un subgrupo de orden $n\text{.}$ Por ejemplo, $A_4$ tiene orden $12$ pero no posee un subgrupo de orden $6\text{.}$ Sin embargo, los Teoremas de Sylow proporcionan un parcial converse para el Teorema de Lagrange, en ciertos casos nos garantizan subgrupos de órdenes específicas. Estos teoremas arrojan un poderoso conjunto de herramientas para la clasificación de todos los grupos finitos nonabelianos.

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