I have the following exercise:
Find all normal non-trivial subgroups of $G_1 \times G_2$. I managed to see that using projections, the normal subgroups are isomorphic to $G_1$ or $G_2$. But the professor say that we need to separate the case when $G_1\cong G_2$. But I don't understand why. Can you help me with this?
You forgot the trivial group and the whole group, for starters.
Using projections, if $N\triangleleft G_1\times G_2$, then $\pi_1(N)=\{e\}$ or $G_1$, and $\pi_2(N)=\{e\}$ or $G_2$, because $G_1$ and $G_2$ are simple and $\pi_1(N)\triangleleft G_1$, $\pi_2(N)\triangleleft G_2$.
If both projections are trivial, then $N=\{e\}\times\{e\}$. If one projection is trivial and the other is nontrivial, then you get $N=G_1\times\{e\}$ and $N=\{e\}\times G_2$.
But if both projections are non-trivial, then what you have is a subdirect product. The subdirect products are described by Goursat’s Lemma. They correspond to isomorphisms between quotients of $G_1$ and quotients of $G_2$.
If $G_1$ is not isomorphic to $G_2$, then the only quotient of $G_1$ that is isomorphic to a quotient of $G_2$ is the trivial one. Goursat’s Lemma tells you that $N$ in that case is $G_1\times G_2$.
But if $G_1\cong G_2$, then you have other possibilities. Specifically, the missing subdirect products are of the following form: fix an isomorphism $f\colon G_1\to G_2$, and then you get the subgroup $N_f$, given by $$N_f=\{(g_1,g_2)\in G_1\times G_2\mid f(g_1)=g_2\};$$ that is, the “graph” of $f$.
You should check whether these subgroups are normal in $G_1\times G_2$. As a hint, you may want to consider the case of $G_1\cong G_2$ abelian, and $G_1\cong G_2$ nonabelian, separately.
This is why you need to consider the case of $G_1\cong G_2$ separately.
I managed to see that using projections, the normal subgroups are isomorphic to $G_1$ or $G_2$.
This is true if we forget about the trivial subgroup and the entire group $G_1 \times G_2$. However, you're asked to actually find all of the normal subgroups! They are not always just $G_1 \times 1$ and $1 \times G_2$. For example, $\mathbb{Z}/2$ is simple, and $(\mathbb{Z}/2) \times (\mathbb{Z}/2)$ has three distinct normal subgroups isomorphic to $\mathbb{Z}/2$.
