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
April 10th, 2012, 02:34 AM   #1
Member
 
Joined: Feb 2012

Posts: 38
Thanks: 0

Normal subgroups

Hi everyone.

Let be a normal subgroups of of finite index . Show that, if is any subgroup of , then is finite and .

Thanks.
Fernando is offline  
 
April 11th, 2012, 12:40 AM   #2
Senior Member
 
Joined: Mar 2012

Posts: 294
Thanks: 88

Re: Normal subgroups

first, let's note that A?N is normal in A: let a be any element of A, and n be any element of A?N.

then ana^-1 is in A, since all of a,n and a^-1 are in A, and A is a subgroup, thus closed under multiplication.

but since a is also in G (since A is a subgroup of G), and N is normal in G, ana^-1 is in N.

thus ana^-1 is in both A, and N, so is in A?N.

hence we can form the group A/A?N, which has order [A:A?N].

now, since N is normal, AN is a subgroup of G, and by the second isomorphism theorem:

AN/N ? A/A?N. since AN is a subgroup of G containing N, AN/N is a subgroup of G/N,

which therefore must be finite (since G/N is finite, of order n), and have order dividing |G/N|.

hence s = [A:A?N] = [AN:N] = |AN/N| divides |G/N| = [G:N] = n.
Deveno is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
normal, subgroups



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Normal Subgroups BrettsGrad Abstract Algebra 2 February 20th, 2013 08:28 AM
Normal subgroups Fernando Abstract Algebra 2 July 5th, 2012 11:09 AM
Normal Subgroups ejote Abstract Algebra 4 February 1st, 2011 07:30 AM
Normal Subgroups!! donwu777 Abstract Algebra 1 December 10th, 2008 04:03 PM
normal subgroups bjh5138 Abstract Algebra 1 October 7th, 2007 09:20 PM





Copyright © 2019 My Math Forum. All rights reserved.