Let's consider a group $G$ and a subgroup $H$.
We note the quotient group $G/H$
My question is: why this notation and the notion of "quotient" ?
Is the motivation only because from Lagrange theorem for finite groups, where we have $|G/H|=|G|/|H|$ or there are deeper reasons ?
Maybe I am lacking some intuition to see further motivations.
Note: I am by far not an expert of group theory, I only know the basics so I expect an answer following the same logic =)