Group theory/Cosets/Properties/Fact/Proof

Proof

The equivalence between ${}(1)$ and ${}(3)$ (and between ${}(2)$ and ${}(4)$) follows by multiplication with ${}y^{-1}$ and with ${}y$. The equivalence between ${}(3)$ and ${}(4)$ follows by going to the inverse elements. From ${}(1)$ we get ${}(5)$ because of ${}1\in H$. If ${}(5)$ holds, then this means that ${}xh_{1}=yh_{2}$ holds with certain ${}h_{1},h_{2}\in H$. Therefore, ${}x=yh_{2}h_{1}^{-1}$, and ${}(1)$ is satisfied. (4) and (6) are equivalent due to the definition. Since the left cosets are the equivalence classes, the equivalence between (5) and (7) follows.

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