I guess it figures that after talking about congruence in groups, it's only natural that we then find some analogy to quotient rings in group theory. I didn't find the concept hard to understand but the difficult part for me was working through the examples.
In pondering the material, the idea that comes to mind is that we aren't really going to be interested in finite quotient groups since they're just going to be isomorphic to some other finite, simpler group. Infinite quotient groups on the hand are probably a lot more interesting.
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