(Teorema de Iwasawa)
Sea G un grupo que actua sobre $\Omega$ tal que se cumplen:
(i) G es un grupo primitivo
(ii) $G^{\prime} =G$
(iii) Si $\alpha \in \Omega$, $G_{\alpha}$ tiene un subgrupo A que es abeliano y normal de modo que
$G=<A^g \mid g\in G>$.
Entonces $G/K$ es un grupo simple.
Preguntas:
1)¿Con este teorema se puede probar que el grupo alternante $A_n$ es simple para $5\leq n$ ?
2) ¿ Para qué otros grupos se usa el teorema para probar que sean simples ?
3) Libros para las aplicaciones de este teorema.
Muchas gracias.
(Iwasawa theorem)
Let G be a group that acts on $\Omega$ such that they are met:
(i) G is a primitive group
(ii) $G^{\prime}=G$
(iii) If $\alpha\in \Omega$, $G_α$ has a subgroup A that is abelian and normal so that $G=<A^g \mid g \in G>$.
So $G /K$ is a simple group.
Questions:
1) With this theorem can you prove that the alternating group $ A_n $ is simple for $5 \leq n$?.
2) For what other groups is the theorem used to prove that they are simple?.
3) Books for the applications of this theorem.
Thank you
Iwasawa's theorem is used in the simplicity proofs of all of the classicial groups (linear, symplectic, unitary and orthogonal) over fields. These proofs generally work for both finite and infinite fields, and the exceptional cases of non-simplicity are generally for small finite fields.
Detailed proofs can be found in the book "The geometry of the Classical groups" by D. E. taylor, and in less detail in "The Finite Simple Groups" by R. A. Wilson. There are also detailed proofs for the linear symplectic and unitary groups (but not for the orthogonal groups) in Huppert's in Chapeter II of Huppert's "Endliche Gruppen I".