= Normal subgroup
{wiki}
Ultimate explanation: https://math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426\#3732426
Only normal subgroups can be used to form <quotient groups>: their key definition is that they plus their cosets form a group.
Intuition:
* https://math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups
* https://math.stackexchange.com/questions/1014535/is-there-any-intuitive-understanding-of-normal-subgroup/1014791
One key intuition is that "a normal subgroup is the <kernel (algebra)>" 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.
Therefore "there aren't that many <group homomorphism>", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.