Let H be a subgroup of a finite group G.
Now we define an equivalence relation a~b if and only if ab^(-1) is in H.
So we can make the partition of G with this equivalence relation.
For each partition is a right coset of H, Ha = {ha : h in H}
Check:
For any h1a,h2a in Ha,h1a(h2a)^(-1)=h1aa^(-1)h2^(-1)= h1h2^(-1) is in H, so h1a and h2a are equivalence. So for each right coset is a equivalence class.
And it is easy to see that the number of each equivalence classes (|Ha|) is equal to the number of H (|H|). By there exist the function f is one to one and onto from H to Ha where the function f(h)=ha.
So |G| = |G / H|*|H|, where the G / H := { all equivalence classes in G }.
complete the proof.
