How can I prove that the order of the group $G$ is equal to the order of each coset multiplied by the number of cosets?

Question

Suppose you have a group $G$ with some subgroup $H$. How can I show that the order of $G$ is equal to the order of each coset of $H$ in $G$ multiplied by the number of different cosets?

Thanks

Answer 1

Hint:

Step 1, show that two different cosets are disjoint so the cosets form a partition of $G$.

Step 2, show that any coset is such that $|xH|=|H|$.

Step 3: $|G|=n_H\times |H|$ where $n_H$ is the number of cosets.

Answer 2

You just have to prove the mappings $H\to gH, h\mapsto gh$, are bijections from $H$ onto cosets. Hence each coset has the same cardinal as $H$. Then prove the cosets modulo $H$ are a partition of $G$.