There is a lemma that says that all left cosets $aH$ of a subgroup $H$ of a group $G$ have the same order.
The proof given is as follows…
The multiplication by $a \in G$ defines the map $H \rightarrow aH$ that sends $h\mapsto ah$. This map is bijective because its inverse is multiplication by $a^{-1}$.
I don't quite understand the proof. Why does having a bijective map mean that all sets of left cosets have the same order?
Thank you
Best Answer
Because this is the definition of having the same cardinality.