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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 .

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