Does this property of normal subgroups have a name? I've recently started studying abstract algebra and I noticed that a subgroup N is normal iff $a \equiv x$ and $b \equiv y$ implies  $ab \equiv xy$ mod N for any a,b,x,y.
Is this an important property and does it have a name I could look up to find out more?
 A: Indeed, normality ($aNa^{-1}\subseteq N$ for all $a\in G$) is actually an easy-to-check condition equivalent to the binary relation $x\equiv y$ defined by $xN=yN$ being a so-called congruence relation, i.e. the equivalence class (here, the coset) of any product $xy$ is determined by the classes (cosets) of $x$ and $y$. Congruence relations can be defined for other algebraic structures and they are indeed important. For instance, an analogous version holds for ideals as subsets of rings.
Ultimately, congruence relations allow us to define quotient structures, such as quotient groups or quotient rings. The first isomorphism theorem (for whatever category one's working with) will tell us that all homomorphic images are isomorphic to quotients. For a quotient, one turns the equivalence classes themselves into elements of an algebraic structure. One way of viewing this is as a "collapsed" structure, where we've imposed new equalities between elements that didn't exist before, and after the dust settles (i.e. one deduces further equalities from the ones we imposed) we get a new structure. Another way of viewing them is as the same structure but at a "lower resolution," where we purposely weaken our eyesight to the point that we can't distinguish certain elements from each other (namely, all elements that are within a given equivalence class are blurred together).
