I have occasionally heard mathematicians speak of "quotienting" as if it were a verb. When someone treats "quotient" (in the context of group theory) as a verb, are they just referring to the natural projection homomorphism $\pi:G \to G/N$ where $N$ is normal in $G$? I obviously get the reasoning for the notion of a "quotient group" since you are just partitioning the group $G$ so it makes sense to me that one could treat "quotient" as a verb. For example, in the construction of the Vitali set, would it make sense to say something like "quotienting $\mathbb{R}$ by $\mathbb{Q}$" or if the context is obvious just "quotient $\mathbb{R}$" etc.?
Is "quotienting" just the operation of taking of taking left cosets of a normal subgroup?
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1Yes, let $G$ be a group and $S\subseteq G$. Quotienting $G$ by $S$ means to (1) construct $N$, the smallest normal subgroup of $G$ containing $S$ and (2) consider $G/N$. In general contexts, extend $S$ to be the smallest object containing $S$ for which a quotient exists and take that quotient. – 2017-01-09
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1The Vitali set is exactly the quotient you mention - at least as an abelian group. The topological structure on it is another story. – 2017-01-09