|Groups|

-September 11, 2013September 11, 2013Uncategorized

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 $o(ST)=\frac{o(S)o(T)}{o(S\cap T)}$ .