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!