Quotienting a group by an equivalence relation In Jacobson's book, Basic Algebra 1, he points out that you can form quotient groups by looking at the equivalence classes of a 'multiplication compatible' equivalence relation on the group. By multiplication compatible, he means that if $A,B$ are equivalence classes for $a, b$, and we also have that $c \in A, d \in B$, then the equivalence class of $ab$ is the equivalence class of $cd$. As it turns out, this forms a group. He then points out that the equivalence class of 1 is a normal subgroup and that the group that you obtained is equal to the original group quotiented by the equivalence class of 1. I'm curious as to if there are any situations where you would quotient a group by an equivalence relation, and not by a normal subgroup, given that these are equivalent.  
 A: Sure. Consider the quotient of a topological group $G$ by the relation "lies in the same connected component." You get a group $\pi_0(G)$ of connected components. This is equivalent to quotienting by the connected component of the identity (which incidentally proves that this subgroup is normal) but I think the equivalence relation is the more natural thing to think about, even if thinking about the normal subgroup is convenient. 
A: One type of equivalence relation one can define on group elements is a double coset.
Another type of equivalence relation you see in group theory has to do with pairs of subgroups, rather than elements.  If $1\leq M\trianglelefteq H \leq G$, then $(H,M)$ is referred to as a pair if $H/M$ is cyclic.  $(H,M)$ is called "good" if $[g,H\cap g H g^{-1}]\not\subseteq M$ for any $g\in G\setminus H$.  Lastly, if $(K,L)$ is another good pair, we call $(H,M)$ and $(K,L)$ related in G if there is a $k\in G$ so that $k^{-1}Hk\cap L=K\cap k^{-1}Mk$.  (There are character theoretic definitions for all of these, too.)  As it turns out, being "related" is an equivalence relation on the set of good pairs in G.
I don't know whether the second example is what you were looking for exactly when you talk about taking a quotient, but the interactions between these equivalency classes can be very interesting, and ends up motivating the study of a complicated class of groups called $M$-groups.
