I'm having a debate with a friend. Let $G$ denote an abelian group, and suppose $A$ and $B$ are subgroups of $G$. The question is whether $(G/A)/B$ makes sense with the standard definitions.
My friend thinks that $(G/A)/B$ does make sense. In his view, when we unpack the definition of quotient group, we find that the elements of this group are of the form $\{(gA)B : g \in G\}$, and hence that it's isomorphic to $G/AB$.
However I think that $(G/A)/B$ does not make sense (with the standard definitions). My argument is that since $B$ is a subgroup of $G$ and not $G/A$, hence we don't actually know what the notation $(G/A)/B$ even means, because this notation is only defined when the denominator is a subgroup of the numerator. To fix the problem, we have to view $G/A$ not as a group, but specifically as a quotient group of $G$, like so:
Definition. Let $G$ denote a group. Then a quotient group of $G$ is a group $Q$ together with a surjective homomorphism $\overline{Q} : G \rightarrow Q$.
Definition. Let $G$ denote an abelian group, $Q$ denote a quotient group of $G$, and $A$ denote a subgroup. Then $Q/B$ is the group defined by $G/(\overline{Q}^{-1}(1) \cdot B)$. We can view $Q/B$ as a quotient group of $G$ by defining $\overline{Q/B}(g) = g\overline{Q}^{-1}(1)B$
With these definitions, I think $(G/A)/B$ makes sense - but you have to regarded $G/A$ as a quotient group of $G$, not just a group.
Question. Who is correct in this debate, and why?
Answer: you are correct, but it is of the 'technically correct' variety. Mathematics is full of notations that do not technically make sense but are acceptable because they only have one interpretation that does make sense. This seems a perfect example of this. I think this phenomenon has been discussed around the web; I'm too lazy to google it now but from memory I recall that Terry Tao's notion of 'post rigorous' and nlab's discussion of 'red herring' both relate to this.
Further discussion: a key difference in my personal opinion between definitions-that-make-sense on one hand and definitions-that-do-not-technically-make-sense-but-are-acceptable-because-they-only-have-one-interpretation-that-does on the other is that the latter type makes more assumptions about the person you are talking to when using such a definition. Concretely you assume
To illustrate 2: I have absolutely no problems with someone talking about $(G/A)/B$ in the context of abelian groups, but would be a bit uneasy if they started doing it in a context where $G, A, B$ are allowed to be non-abelian. Other people would perhaps not feel uneasy at all there, where yet others (you, it seems) are already uneasy in the Abelian case.
To illustrate an other issue with the 'and no others' of 1: my way of making sense of the definition would be somewhat different from yours, and it is non-trivial that they are equivalent. The first part (expressed in your first definition) we agree on: we need to think of a quotient group not only as the resulting group, but as that group together with the quotient map $\overline{Q}$. Thanks for putting that so clearly. But in the case $Q = G/A$ discussed here I would define the 'ill-defined' concept $Q/B$ we are trying to define here simply as $Q/\overline{Q}(B)$ and not as a quotient of the original group as you do.
In the end it is the same thing, but the fact that this equivalence between interpretations is itself non-trivival could be construed as another 'problem' with the notion of $(G/A)/B$ if you will.