Cosets, Lagrange’s theorem and normal subgroups

Our goal will be to generalize the construction of the group Z/nZ. The idea there was to start with the group Z and the subgroup nZ = 〈n〉, where n ∈ N, and to construct a set Z/nZ which then turned out to be a group (under addition) as well. (There are two binary operations + and · on Z, but Z is just a group under addition. Thus, the fact that we can also define multiplication on Z/nZ will not play a role here, but its natural generalization is very important in Modern Algebra II.) We would like to generalize the above constructions, beginning with congruence mod n, to the case of a general group G (written multiplicatively) together with a subgroup H of G. However, we will have to be very careful if G is not abelian.