|HK|=|H||K|/|H intersect K|

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This post is about how to prove that |HK|=\frac{|H||K|}{|H\cap K|}, where H and K are finite subgroups of a group G.

A tempting thing to do is to use the “Second Isomorphism Theorem”, HK/H\cong K/(H\cap K). However that would be a serious mistake since the conditions for the Second Isomorphism Theorem are not met. In fact HK may not even be a group.

The correct way is to note that HK=\bigcup_{h\in H}hK.

Therefore |HK|=|K|\times |\{hK:h\in H\}|. For h_1,h_2\in H, we have:

\begin{aligned}h_1K=h_2K&\iff h_1h_2^{-1}\in K\\    &\iff h_1h_2^{-1}\in H\cap K\\    &\iff h_1(H\cap K)=h_2(H\cap K)    \end{aligned}

Therefore |\{hK:h\in H\}|=|\{h(H\cap K):h\in H\}|, i.e. the number of distinct cosets h(H\cap K). Since H\cap K is a subgroup of H, applying Lagrange’s Theorem gives the number of distinct cosets h(H\cap K) to be \frac{|H|}{|H\cap K|}.

Thus, we have |HK|=\frac{|H|}{|H\cap K|}\cdot |K|.


Undergraduate Math Books

 

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