I want to know the answer to the following question:
What group does the inverse of gN include?
All the background I'm provided with is that "This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties.". Is that enough to answer the question?
Answer: no