by ayushkhaitan3437

Today we will discuss the proof of o(ST)=\frac{o(S)o(T)}{o(S\cap T)}.
Here, S and T are groups. We know S\cap T\neq\emptyset, as e\in S\cap T.

Let s_1t_1=s_2t_2. Then s_1s_2^{-1}=t_2t_1^{-1}\in S\cap T. Take any a\in S\cap T. For any s_1,t_1\in S,T, find s_2=s_1a^{-1} and t_2=at_1. Then s_2t_2=s_1a^{-1}at_1=s_1t_1. Hence, |S\cap T| pairs of elements (s_2,t_2) can be found such that s_2t_2=s_1t_1 for any two s_1,t_1\in S,T. Hence, we can form equivalence classes which partition ST, all with |S\cap T| elements. This shows o(ST)=\frac{o(S)o(T)}{o(S\cap T)}.

We can also digress to more complicated situations like o(ST+W), and find similar formulae.