group-theory
IN Herstein has stated in the books Topics in Algebra that a proof of the above conjecture was given by Walter Feit and John Thompson.(he defines a simple group as one which has no-nontrivial homomorphic images).
Question: Can anyone please show me a proof of the above conjecture using the group theory background in Herstein's text or at least one I can understand?I am currently studying about homomorphisms from there.
Thanks!
Rotman's Introduction to the Theory of Groups (theorem 6.9 on page 130) contains such a proof, attributed to Schenkman, but this proof is not exactly as I have outlined in the question.
The book The Theory of Finite Groups: An Introduction, by Kurzweil and Stellmacher, does what I had in mind in Chapter 2.
This is basically the First Ismorphism Theorem. Let's say $H$ is a homomorphic image of $G$, i.e. there exists a morphism $\varphi$ such that $\varphi : G \to H$ is surjective. Then $N:= \text{ker} \ \varphi$ is a normal subgroup of $G$ and $G/N \cong H$. Then the correspondence \begin{align} \{ \text{normal subgroups of } G \} &\to \{ \text{homomorphic image of } G \} \\ N &\mapsto G/N \end{align} is inverse to (up to isomorphism) \begin{align} \{ \text{homomorphic image of } G \} &\to \{ \text{normal subgroups of } G \} \\ H &\mapsto N:= \text{ker} \ \varphi, \text{ where } \varphi : G \to H. \end{align}
Best Answer
This isn't at all an answer to your question, but as you've already seen, you're not going to get one!
I thought you might be interested to know that there is now a formally verified computer proof of the Feit-Thompson Theorem. It took six years to produce and was completed just last year. Here's an article about it.