On the notation “$G/N$” for quotient groups : Abstract Algebra (3rd Edition), Dummit and Foote, Pg 82

Question

Problem from Abstract Algebra (3rd Edition), by Dummit and Foote, Chapter 3, Exercise 3.1, pg 85

This question has been answered before on StackExchange but a consensus was not realized as many of those who answered this question did not possess the book (Use of the symbol $G/N$ for the quotient group).

I quote, verbatim, the following from the above book:

"Proposition 7. A subgroup N of the group G is normal if and only if it is the kernel of
some homomorphism."

My concern is the following: Up until this point, the quotient group is defined to G/K, where K is the kernel of some homomorphism from G to H. But in order to prove that if a subgroup N of G is normal, then it will necessarily be the kernel of some homomorphism between G to some group "H" the author(s) state that this group "H" would be of form G/N.

BUT how can the authors say that this group would be of the form G/N without first proving that N is the kernel of some homomorphism? (In other words, how can we write G/N is a valid object until and unless we prove N is indeed the kernel?)

Furthermore, the authors also write in page 82, about five paragraphs above Proposition 7 "Note that the structure of G is reflected in the structure of the quotient G/N when N is a normal subgroup (for example, the associativity of the multiplication in GIN is induced from the associativity in G and inverses in G 1 N are induced from inverses in G)." And I am again troubled by the G/N notation. How can, with all due respect, the author(s) invoke this notation without first proving that N is indeed the kernel of some homomorphism?

What am I missing? Or is the G/N is just an abuse of notation to represent the the left cosets of N in G? And we write G/N because we have the foresight that N would indeed be the kernel of some homomorphism?

Answer 1

I am looking at my copy of Dummit & Foote (second edition). Proposition 7 appears on page 83 at the bottom, so we are clearly looking at different editions. That said...

Just before Proposition 4, the authors say:

By Theorem 3, if we are given a subgroup $K$ of a group $G$ which we know is the kernel of some homomorphism, we may define the quotient $G/K$ without recourse to the homomorphism by the multiplication $uKvK = uvK$. This raises the question of whether it is possible to define the quotient group $G/N$ for any subgroup $N$ of $G$. The answer is no in general since this multiplication is not in general well defined (cf. Propositon 5 later).

Then after Proposition 5, at the bottom of Page 82, they state:

As indicated before, the subgroups $N$ satisfying the condition in Propositio 5 for which there is a natural group structure on the quotient $G/N$ are given a name:

That is, the authors have already established that for normal subgroups $N$, the set of cosets, which they are already denoting $G/N$ in previous paragraphs, have a group structure given by the multiplication $uNvN = uvN$. This happens before Proposition 7. They have indicated you can consider the set of cosets for any subgroup (on pp. 81 before Proposition 4), and have established that the multiplication given that way is well-defined and yields a group if and only if the subgroup $N$ satisfies $gng^{-1}\in N$ for all $g\in G$ and all $n\in N$ (that is, $N\triangleleft G$), in Proposition 5. So they can certainly use that notation in Proposition 7, having already introduced it and shown it yields a group whenever $N$ is a normal subgroup.