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1. Let G={invertible upper-triangular 2x2 matrices}

H={invertbile diagonal matrices}

K={upper-triangular matrices with diagonal entries 1}

We are supposed to determine if G is isomorphic to the product of H and K. I have concluded that this is true, but I'm having trouble proving it.

I need to show three things: (1)H,K are normal subgroups of G; (2)the intersection of H and K is the identity; (3)HK=G

I can do (3), I can see that (2) is true (though I haven't written it up formally yet), and I can show that K is a normal subgroup of G.

I'm having trouble showing H is a normal subgroup of G. I tried finding a homomorphism between G and K and showing the kernal of that homomorphism is H (this is how I showed K is normal), but I can't find a function that works. I've also tried showing that there are elments g in G and h in H such that ghg[tex]^-^1[/tex] is in H, but to no avail. So I'm stuck.

2. Prove that the product of two infinite cyclic groups is not infinite cyclic.

So far, this is what I have:

Let H and G be two infinite cyclic groups. Let H be generated by h and G by g. Also, the product of G and H, GxH={(g,h) such that g is in G and h in H}. So I need to show there is no element (a,b) in GxH suth that (a,b) generates GxH. That is, show there is no (a,b) in GxH such that (a[tex]^n[/tex],b[tex]^n[/tex])=(g[tex]^i[/tex],h[tex]^j[/tex]) for any i,j. This is where I'm stuck.