1. ## Normal subgroup question

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!

2. Elements of $G_1 \times G_2$ are pairs of elements of G1 and G2, while elements of $N_1 \times N_2$ are pairs of elements of N1 and N2.
If $g \in G_1 \times G_2$, and $h \in N_1 \times N_2$ how can you write them in terms of each group? How is group multiplication defined in $G_1 \times G_2$? What can you then say about $g^{-1}hg$?

3. Here was my outline for doing the whole problem:

$N_1 \times N_2 \triangleright G_1 \times G_2 \leftrightarrow (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

So, to show that $N_1 \times N_2 \triangleright G_1 \times G_2$, I want to show that $(g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

[If I had $N_1 \times Z_1 \triangleright G_1 \times G_2$ and $N_1 \times Z_1 \cong N_1$, then $(g_1g_2)(n_1)(g_1g_2)^{-1} \in N_1$]

Similarly,

[If I had $N_2 \times Z_1 \triangleright G_1 \times G_2$ and $N_2 \times Z_1 \cong N_2$, then $(g_1g_2)(n_2)(g_1g_2)^{-1} \in N_2$]

Then by looking at $N_1 \times N_2$, I would multiply the previous results to obtain this:

$(g_1g_2)(n_1)(g_1g_2)^{-1}(g_1g_2)(n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

$= (g_1g_2)(n_1)g_2^{-1}g_1^{-1}g_1g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

$= (g_1g_2)(n_1)g_2^{-1}g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

$= (g_1g_2)(n_1n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

$= (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$ QED.

Is this an acceptable approach? The material in the colored brackets is what I'm not comfortable assuming.