Ciro Santilli
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Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.
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One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.
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Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
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The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.
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Ancestors

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