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

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 its soluble by FeitThompson). 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?

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

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

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.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,242

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

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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