Elements of are pairs of elements of G1 and G2, while elements of are pairs of elements of N1 and N2.
If , and how can you write them in terms of each group? How is group multiplication defined in ? What can you then say about ?
Hi, everyone! This is my first post on the forums.
There is a practice problem I've found that states:
"Given N1, a normal subgroup of G1, and N2, a normal subgroup of G2, show N1 X N2 is a normal subgroup of G1 X G2."
Now I've been doing a lot of approaches, but the one that seems to be working out for me hinges on if I can assume that N1 is a normal subgroup of G1 X G2 (which then means I can assume as well that N2 is a normal subgroup of G1 X G2). But can I make this assumption? Is it true? Why or why not?
EDIT: I realize that I should be working with N1 X Z1 and N2 X Z1 above so that they can be normal subgroups (which are isomorphic to N1 and N2 respectively). I'm still unsure of how to show that it is the case that N1 X Z1 is a normal subgroup of G1 X G2 and N2 X Z1 is a normal subgroup of G1 X G2.
Thanks for any input!
Here was my outline for doing the whole problem:
So, to show that , I want to show that
[If I had and , then ]
Similarly,
[If I had and , then ]
Then by looking at , I would multiply the previous results to obtain this:
QED.
Is this an acceptable approach? The material in the colored brackets is what I'm not comfortable assuming.