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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.

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up vote 2 down vote accepted

You can also write $H$ as $G \times \{e\}$. This is the kernel of the projection onto the second factor, i.e. the homomorphism $G \times G \to G$ sending $(x, y)$ to $y$. It isn't so hard to check normality directly, either: take an element $(x, e) \in H$ and conjugate by some $(y, z)$. What happens?

[If $g$ means what I suspect it does, then your $f$ is nearly identical to mine and has the same kernel: you're just sending $(x, y)$ to $(e, y)$.]

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Oops, yeah I guess I meant $f(x_1,y_1)=(x_1,y_1)(x_1^{-1},e_G)$. Got it now, thanks again. – user2467 Feb 12 '12 at 7:25

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