My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 29th, 2007, 11:03 PM   #1
Newbie
 
Joined: Feb 2007

Posts: 8
Thanks: 0

Finitely Generated

Suppose G is finitley generated group for every n prove that there are finite subgroups like H such that [G:H]=n
payman_pm is offline  
 
March 30th, 2007, 04:11 AM   #2
Member
 
Joined: Dec 2006

Posts: 39
Thanks: 0

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
Tonio is offline  
March 30th, 2007, 05:58 AM   #3
Site Founder
 
julien's Avatar
 
Joined: Nov 2006
From: France

Posts: 824
Thanks: 7

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 ...
julien is offline  
March 30th, 2007, 10:20 AM   #4
Newbie
 
Joined: Feb 2007

Posts: 8
Thanks: 0

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.
payman_pm is offline  
March 31st, 2007, 09:54 AM   #5
Member
 
Joined: Dec 2006

Posts: 39
Thanks: 0

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.
********************************
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
Tonio is offline  
July 22nd, 2007, 01:02 AM   #6
Newbie
 
Joined: Jul 2007

Posts: 1
Thanks: 0

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.
i_love_galois is offline  
August 13th, 2007, 01:56 AM   #7
Member
 
Joined: Dec 2006

Posts: 39
Thanks: 0

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
Tonio is offline  
September 19th, 2007, 04:21 AM   #8
Newbie
 
Joined: Sep 2007

Posts: 3
Thanks: 0

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
yeela is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
finitely, generated



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Prove S is not finitely generated. please help kennedy Linear Algebra 1 October 1st, 2009 03:59 PM
Noetherian, finitely generated R-module poincare4223 Abstract Algebra 0 April 29th, 2009 09:01 PM
finitely generated ideal, idempotent mingcai6172 Abstract Algebra 0 April 29th, 2009 08:42 PM
T-groups finitely generated sastra81 Abstract Algebra 0 January 3rd, 2007 06:54 AM
question about the groups finitely generated sastra81 Abstract Algebra 1 January 2nd, 2007 12:03 PM





Copyright © 2018 My Math Forum. All rights reserved.