Finally, the LaTeX path not specified problem has been solved by WordPress!
This post is about how to prove that , where and are finite subgroups of a group .
A tempting thing to do is to use the “Second Isomorphism Theorem”, . However that would be a serious mistake since the conditions for the Second Isomorphism Theorem are not met. In fact may not even be a group.
The correct way is to note that .
Therefore . For , we have:
Therefore , i.e. the number of distinct cosets . Since is a subgroup of , applying Lagrange’s Theorem gives the number of distinct cosets to be .
Thus, we have .