How can groups be classified—is it even possible, and by what criteria? To answer these legitimate questions, we must first examine them and understand their inner workings. So let us take out the algebraist’s toolkit and consider an arbitrary group G, not necessarily finite, equipped with an operation that need not be commutative and that we shall write multiplicatively. What do we immediately see in G? Scattered elements that we may choose to gather into subsets. A subset H of G may itself be a group; a necessary condition is that it contain the identity element e of the operation. It is then called a subgroup of G. Are there any particular relationships between elements of G, both inside and outside H?
A class act! -----------------
Focus on an element a of G that does not belong to H. For every element h of H, the product a.h then lies outside the subgroup H. To see this, suppose for a contradiction that a.h belongs to H. There is then some h’ in H such that a.h = h’. Since H is itself a group, the inverse h−1 of h also belongs to H, which is moreover closed under its internal operation. It follows that h’.h−1 belongs to H. But h’.h−1 equals a.h.h−1, and hence a, which was chosen outside H. Contradiction!
This elementary observation is the key to organizing groups. The set of elements of H multiplied on the left by a plays an important role; we shall denote it by aH. Thus, aH = { g ∈\in G, g = a.h, h ∈\in H }. We have seen that aH ∩\cap H = ∅\varnothing: the intersection of H and aH is empty.
To become familiar with this notation, observe that for every element h0 of H, we have h0H = H, since H is a group. In particular, eH = H.