Math Is Fun Forum

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).

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?

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.

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"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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: The Feit–Thompson theorem

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.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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.

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"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..."