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April 10th, 2012, 02:34 AM   #1
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Normal subgroups

Hi everyone.

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

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April 11th, 2012, 12:40 AM   #2
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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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