I'm really stuck on these two questions, please help!(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Normal subgroups, isomorphisms, and cyclic groups

**Physics Forums | Science Articles, Homework Help, Discussion**