# isomorphic to product

• Sep 6th 2011, 05:22 PM
shelford
isomorphic to product
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
• Sep 6th 2011, 05:46 PM
Drexel28
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.
• Sep 6th 2011, 06:04 PM
shelford
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