The benefits of first presenting cosets as an equivalence relation instead of a translation In Peter Cameron's Introduction to Algebra the definition of a coset introduced is that produced from an equivalence relation and then it is shown this can be seen as the translation of a subgroup.
In Joseph Rotman's An Introduction to the Theory of Groups the definition of a coset is introduced as the translation of a subgroup and then it is shown these partition the group and so there must be an equivalence relation around somewhere.
My question is why does Cameron do this, or why might he? It seems more natural to take Rotman's approach, especially with the subsequent example of the additive group of $\mathbb{R}^2$ that motivates the idea of a coset. This as opposed to presenting a seemingly out of nowhere equivalence relation.
 A: Cameron first introduces cosets (of subrings) in the context of rings in the previous chapter. His first example is the $4$ cosets of $4\Bbb Z$ in $\Bbb Z$, and he notes that in general the cosets of $n\Bbb Z$ are the congruence classes mod $n$. He immediately goes on to introduce ring homomorphisms and ideals. In this context it is entirely natural to start with the equivalence relation and then note that its classes can also be viewed as translates of the subring. It then makes excellent pædagogical sense to take the same approach in the next chapter when he deals with subgroups and then with group homomorphisms and normal subgroups.
Besides, the technique of introducing an equivalence relation and extracting information from the induced partition or quotient is of wide applicability but tends — at least in my experience — to be overlooked in undergraduate mathematics. Besides, Cameron is a combinatorist, and it’s a natural combinatorial approach.
A: I don’t know Cameron, so won’t speculate about his intentions.
Often in math it is fruitful to consider quotients by some equivalence relation, f.eks so as to discard irrelevant information. In the context of group theory it is natural to ask under which equivalence relations $[x]+[y]=[x+y]$ is well defined. This turns out to be the case exactly when there is a subgroup $A$ such that $x\sim y$ if and only if $y-x\in A$. To me it always seems nice to set forth what you want to achieve, and see if that forces your definitions.
A: I was just thinking about this question, and offer a slightly different take. I want to attempt to rebut the suggestion that the equivalence relation was "seemingly out of nowhere".
If you do $x$, then do $x^{-1}$, you always end up back where you started. This is a good way of defining the inverse: it's the thing that gets you back to where you started. Uniqueness of inverses is the statement that the set of all elements $y$ such that $xy^{-1}\in \langle 1\rangle$ is simply $\{x\}$. This is a rather boring equivalence relation, but it is one.
A sort of generalized inverse then, is where we demand not that $xy^{-1}$ is in $\langle 1\rangle$, but is kept inside a subset, say $H$. Thus $\{y\mid xy^{-1}\in H\}$ becomes a generalization of the idea that inverses are unique, to $H$-generalized inverses are in $1-1$ correspondence with the elements of $H$. If $H$ is an arbitrary set then the $H$-generalized inverses form a set of size $|H|$. One then can see that these sets of generalized inverses are disjoint if and only if $H$ is a subgroup.
