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Obob
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 Re: Lagrange's Theorem   « Reply #25 on: Sep 22nd, 2007, 7:24pm » Quote Modify

Yes, Eigenray, you are right.  I was trying to make the proof as simple as possible, and ended up making it simpler than possible  .

For a real proof, note that any group of order pa has nontrivial center.  This follows from letting G act on itself by conjugation; then all orbits have order a power of p, and since their sum is the order of G and the identity has an orbit of size 1, there must be other orbits of size 1.  Now pick a nonidentity element g from the center.  Some power h of g has order p, and <h> is normal.  So we can consider G/<h>, and proceed by induction.
 « Last Edit: Sep 22nd, 2007, 7:24pm by Obob » IP Logged
ecoist
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 Re: Lagrange's Theorem   « Reply #26 on: Oct 8th, 2007, 2:29pm » Quote Modify

Solution for Eigenray's easier problem:

Determine those integers n such that if n divides the order of a group G, then G has a subgroup of index n.

We show that the only such n is n=1.  Let n>1.  Let G be the alternating group on 2n letters.  Then G is a simple group of order greater than n!, and its order is divisible by n.  If G had a subgroup of index n, then G would be isomorphic to a subgroup of the symmetric group on n letters, contradicting the fact that G has order larger than n!.
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