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I'm sure this question has been asked somewhere, but I have been unable to find it. Why is it that when we have some set $X$ with an equivalence relation $\equiv$, and $X$ has some structure (e.g. a group operation, a topology, a vector space), we refer to the structure induced by $\equiv$ on $X / \equiv$ a "quotient" (group, topology, vector space, etc.). My guess would be that it comes from a prior use of "quotient" to talk about division on $\mathbb{R}$, and so the field of fractions on $\mathbb{Z}$ (with an equivalence relation in the from of $x / y \equiv ax / ay$) was the source of using "quotient". Is this correct? If not, what is the origin behind referring to the induced structure as a "quotient"?

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Probably the origin is with the theory of groups: see quotient group.

According to William Burnside in:

Herr Hölder [footnote: "Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen", Math.Ann.,‎ 1889] has introduced the symbol

$$\dfrac G \Gamma$$

to represent this group; he calls it the quotient of $G$ by $\Gamma$ and a factor-group of $G$.

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