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 payman_pm March 29th, 2007 11:03 PM

Finitely Generated

Suppose G is finitley generated group for every n prove that there are finite subgroups like H such that [G:H]=n

 Tonio March 30th, 2007 04:11 AM

Re: Finitely Generated

Quote:
 Originally Posted by payman_pm Suppose G is finitley generated group for every n prove that there are finite subgroups like H such that [G:H]=n
What do you mean? There must be a period or a comma somewhere there in the middle of that phrase in order to be a clear one.
Now, it is NOT true that if G is a f.g. group then for any n (natural number?) there's a sbgp. of index n. Just take some finite group and let n be a natural number that does NOT divide the order of G...
Tonio

 julien March 30th, 2007 05:58 AM

I tried to add the infiniteness but I am unable even to start. If that kind of problem is not open, there must be some pretty heavy results behind ...

 payman_pm March 30th, 2007 10:20 AM

I mean
G is a group which is finitely generated and n is natural number .prove that there are finite subgroups like H such that [G:H]=n.

 Tonio March 31st, 2007 09:54 AM

Quote:
 Originally Posted by payman_pm I mean G is a group which is finitely generated and n is natural number .prove that there are finite subgroups like H such that [G:H]=n.
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Again, this is NOT true. If G is finite, then there are plenty of counterexamples for plenty of natural numbers....for example, no group of order 36 has a sbgp. of index 7,5, 11 or 23.
Now if G is infinite, then if H is finite the index [G:H] HAS to be infinite, so the question AGAIN makes no sense
Tonio

 i_love_galois July 22nd, 2007 01:02 AM

let G=<a1,...,am>. If |G:H|=n, we can get the generators of H by the coset representatives. We know the are finite many different representatives by Reidemester-Schreier system.

 Tonio August 13th, 2007 01:56 AM

Quote:
 Originally Posted by payman_pm I mean G is a group which is finitely generated and n is natural number .prove that there are finite subgroups like H such that [G:H]=n.
**************************************

Revisiting the site after sometime I think I may, probably, know what the OP meant to ask: he wants to prove that if G is a f.g. group, then for any natural number n there's a finite number of sbgps. H of G s.t. [G:H] = n.
If this is what he meant to ask....wow, dude! Please do improve your english! Anyway you'll need it to make a reasonably worthwhile career in maths since a huge ammount of professional papers and books are in english.

Anyway: Let H <= G be s.t. [G:H] = n ==> by the regular representation of G in the set of left cosets of H in G, we get a homomorphism G --> S_n.
Since any homom. from G to any group is uniquely determined by its action on any set of generators of G, and since G is fin. generated (say, by g1,..., gr), there thus are at most (n!)^r homom. from G into S_n, and since any sbgp. H as above gives rise to one of these, there are at most (n!)^r sbgps. of G of index n.

Regards
Tonio

 yeela September 19th, 2007 04:21 AM

Tonio,
I can't see how the representation you mention uniquely determines H. There might be more than one H giving rise to the same permutation representation.
This is true already for the subgroups of order 2 in Z/2Z x Z/2Z.

/d

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