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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 =)

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    $\begingroup$ Does this answer your question? Who named "Quotient groups"? $\endgroup$ – DatCorno Nov 19 '20 at 15:36
  • $\begingroup$ Do you intend $H$ to be normal in $G$? $\endgroup$ – Shaun Nov 19 '20 at 15:36
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The notation working out nicely for Lagrange's theorem is one reason. There's a couple others:

$\mathbb{Z}/n\mathbb{Z}$ 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

$$\frac{G}{H} \simeq \frac{G/K}{H/K}$$

That is, we can "cancel" subgroups in a quotient group just like we can cancel factors in an actual fraction.

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