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Let $W=X/Y$ be a quotient of $X$

Let $W/Z$ be a quotient of $W$

To write $W/Z$ without $W$, Would we need to write $(X/Y)/Z$? I presume $X/Y/Z$ is ambiguous?

I'm imagining a group quotient and not assuming this is necessarily the same or different for other types of quotient.

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Yes, one would write $(X/Y)/Z$.

However, note that this implies that $Z$ is a normal subgroup of $W$, and thus since $W$ is a quotient group of $X$ we may identify $Z = \pi(K)$ where $K$ is a normal subgroup of $X$, $Y \subseteq K$, $\pi: X \rightarrow X/Y = W$ is the canonical projection. Then by the third iso theorem we have

$$(X/Y) /Z = (X/Y)/(K/Y) = X/K ~~.$$

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  • $\begingroup$ Thanks. Does this generalise beyond groups? e.g. monoids, sets, other objects? $\endgroup$
    – Robert Frost
    Commented Mar 25, 2020 at 10:42
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    $\begingroup$ For rings and modules, this works quite the same. Something similar can be said about universal algebras, though the theorem may not be quite as clean as it is for groups and rings. See here for the most general setting. $\endgroup$
    – G. Chiusole
    Commented Mar 25, 2020 at 10:45

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