/sci/ - Science & Math

Is there a typo in this problem? Shouldn't it be [math]\varphi : G \rightarrow G[/math]?
Because otherwise it seems really simple: [math]H \cap N[/math] must be trivial, which means that [math]\varphi[/math] is injective, so [math]G = H, N = \{1\}[/math], which makes the product map in the question very uninsteresting.
This is from Artin's algebra.