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#1 2009-05-19 04:48:48

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

The Feit–Thompson theorem

Okay, I only just found out what it is. All groups of odd order are soluble (or solvable). smile

And this implies that all finite groups of odd composite order are not simple? I can prove it.


Let
be a finite group of composite odd order (so it’s soluble by Feit–Thompson). We may assume that
is not Abelian, since we all know that any Abelian group of composite order is not simple. Since it is not Abelian, its derived subgroup
is not trvial. Also
cannot be all of
otherwise
for all
whereas being soluble means that
must be trivial after a finite number of derivations. Hence the commutator subgroup of
is a nontrivial and proper normal subgroup, proving that
is not simple.

There. That was not really heavy machinery, was it? tongue


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#2 2009-05-19 05:06:40

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: The Feit–Thompson theorem

There. That was not really heavy machinery, was it?

That solvability of composite order implies nonsimplicity is not, no.  In fact, it becomes more obvious when you use the decomposition series that comes from being solvable.  On the other hand, Feit-Thompson's theorem required 255 pages of heavy machinery to prove.  I'm not sure of any proofs discovered since then, so perhaps it has become shorter with time.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2009-05-19 05:14:34

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: The Feit–Thompson theorem

Ricky wrote:

On the other hand, Feit-Thompson's theorem required 255 pages of heavy machinery to prove.

Oh, that puts a different complexion on the matter then. eek


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#4 2009-05-19 07:54:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,443

Re: The Feit–Thompson theorem

Thought I read about a 1000 page proof in group theory. Andy's is 200, this is 250. What ever happened to the Greeks drive for simplicity?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2009-05-19 08:44:07

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: The Feit–Thompson theorem

The first half of the Jordan-Holder program is estimated at 10,000 pages.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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