Let $G$ be a group and let $H:=\{(g,e_G) \ | \ g \in G\}$. Is $H$ normal in $G \times G$?
I first tried to find a counterexample (picked $S_3$ for $G$), but all of the cosets I checked ended up being the same. So then I tried proving it was normal, but I got stuck. In particular, I tried to show that $H_1$ was the kernel of a homomorphism. I defined $f: G \times G \rightarrow G \times G$ by $f(x)=x*(g^{-1},e_G)$, but I couldn't show that this was a homomorphism (probably because it's not one).
Any guidance would be appreciated.