### |Groups|

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.