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# Group Theory Watch

1. I am given that G1, G2 are finite and K is a subgroup of G1*G2

Also H1={g in G1 s.t. (g,e) in K) and H2={g in G2 s.t. (g,e) in K)

I also have been given that |G1| and |G2| are coprime

I have already shown that H1 is a subgroup of G1 and H2 is a subgroup of G2 and H1*H2 is a subgroup of K.

I need to show that K=H1*H2.

I tried to turn this into a pure number theory problem, lettting |H1|=h1 etc so

h1|g1, h2|g2, h1*h2|k, g1 and g2 are coprime

but this was not enough to show h1*h2=k (which is what I need).
2. You need to show is a subgroup of which combined with the fact is a subgroup of will give you that
3. (Original post by Noble.)
You need to show is a subgroup of which combined with the fact is a subgroup of will give you that
How can I show that K is a subgroup of H1*H2?
4. Let be arbitrary, you want to show that and

Let then as is closed under multiplication

Also gives you that

and

and

However, since this implies

Then if you let by Bezout's there exists integers such that

Now, you want to use Bezout's to show that implies and implies

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Updated: April 12, 2013
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