Re: isomorphic to product

Quote:

Originally Posted by

**shelford** Hi, if someone can explain how to get the answer to this is would be greatly appreciated.

Let:

G be the set of matrices of the form:

[a b : 0 1]

H be the set of matrices of the form:

[a 0 : 0 1]

K be the set of matrices of the form:

[1 b : 0 1]

Q1: Is G isomorphic to H x K? Prove it, or give a counterexample.

Q2: Is G/K isomorphic to H? Prove it, or give a counterexample.

Any help would be nice.

Thanks

What have you tried? Please show some work.

Re: isomorphic to product

For 1: to show G is isomorphic to H x K I need to show:

i) G=HK

ii) $\displaystyle H \cap K = I$

ii) H, K are normal subgroups of G

for i) HK=[a ab : 0 1]

ii) $\displaystyle H \cap K = \{ 1 \}$

Not sure if this is correct and where to go now.

How do I prove that H and K are normal subgroups of G?

And I don't know how to do q2 at all.

Thanks