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. 
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. 

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