ABSTRACT
Recall that the group operation in the additive group ℤ n is given by [a] + [b] = [a + b] where [a] stands for the residue class a + nℤ of a modulo n. Now observe that U ≔ nℤ is a subgroup of ℤ and that [a] = a + U is just a “translate”, i.e., a coset of U in ℤ. Hence the elements of ℤ n are exactly the cosets of U in ℤ, and the group operation is given by (a +U) + (b + U) = (a + b) + U. Maybe we can take an arbitrary group G, a subgroup U ≤ G and then get a group operation on the coset space G/U by (aU) ∘ (bU)≔ (ab)U!?
