Reason of notation for quotient group [duplicate]

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 =)

• Does this answer your question? Who named "Quotient groups"? – DatCorno Nov 19 '20 at 15:36
• Do you intend $H$ to be normal in $G$? – Shaun Nov 19 '20 at 15:36

$$\mathbb{Z}/n\mathbb{Z}$$ is our prototypical case for quotient groups, and here it really is very closely related to division; the equivalence class of $$x$$ mod $$n$$ is the (equivalence class of the) remainder of $$x$$ divided by $$n$$.
The notation also works out very nicely for the third isomorphism theorem (which, if you haven't learned it yet, you will shortly). Namely, if $$G$$ is a group, and $$K < H < G$$, with everything normal in everything else, then
$$\frac{G}{H} \simeq \frac{G/K}{H/K}$$