For the last few weeks I’ve been working as a counselor at the PROMYS program. The program runs, among other things, a course in abstract algebra, which was a good opportunity for me to get annoyed at the way people normally introduce normal subgroups, which is via the following unmotivated
Definition: A subgroup of a group is normal if for all .
It is then proven that normal subgroups are precisely the kernels of surjective group homomorphisms . In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.
Today I’d like to present an alternate conceptual route to this definition starting from equivalence relations and quotients. This route also leads to ideals in rings and, among other things, highlights the special role of the existence of inverses in the theory of groups and rings (in the latter I mean additive inverses). The categorical setting for this discussion is the notion of a kernel pair and of an internal equivalence relation in a category, but for the sake of accessibility we will not use this language explicitly.
Equivalence relations and quotients of sets
Let be a set and let be a function. We can think of as capturing some property of the objects of with values in . If is not injective, then doesn’t completely capture all properties of elements of , but it does capture something. What exactly does it capture?
Consider the relation . This relation inherits the following properties from the properties of ordinary equality:
- reflexivity: ,
- transitivity: ,
- symmetry: .
A relation on a set with these properties is an equivalence relation, and the above axioms are in fact enough to describe precisely the relations we can get in this way. To see this, note that an equivalence relation on a set partitions it into disjoint subsets, the equivalence classes, which consist of maximal collections of elements which are equivalent to each other. We may write the set of equivalence classes as , and then there is a canonical surjective function
assigning an element of its equivalence class. Moreover, the procedure above which constructs an equivalence relation from a function outputs here. Thus talking about surjective functions out of (or quotient sets of ), up to a suitable notion of isomorphism, is equivalent to talking about equivalence relations on , and what a given surjective function on captures is precisely the equivalence class an element of belongs to.
Equivalence relations and quotients of groups
Let be a group and let be a group homomorphism. As before, the relation is an equivalence relation on . However, because is also a group homomorphism, if and then . This gives an additional axiom:
- compatibility with multiplication: if and then .
This defines, in the terminology of universal algebra, a congruence relation on groups. Note that compatibility with multiplication, in the presence of the other axioms defining an equivalence relation, is equivalent to the condition that the equivalence relation, as a subset
of , is a subgroup.
Compatibility with multiplication is precisely the condition needed for multiplication in to be well-defined on the equivalence classes , so given a congruence relation on a group we can recover a quotient map which is a group homomorphism. However, due to inverses we can say more. Compatibility with multiplication shows that
In other words, a congruence relation is completely determined by which elements are congruent to the identity; call these elements . (The corresponding relation might be called congruence by analogy with the case of .) Then:
- having an identity is equivalent to being reflexive,
- being closed under multiplication is equivalent to being transitive, and
- being closed under inverses is equivalent to being symmetric.
In other words, is an equivalence relation if and only if is a subgroup. This is fairly special to groups; it highlights a close relation between groups, group actions, and equivalence relations which motivates the definition of a groupoid.
But we also want compatibility under multiplication, and since
it follows that has another property: it is closed under conjugation, so it is a normal subgroup. Conversely, if we define an equivalence relation by where is a normal subgroup, then
so normality of is equivalent to being compatible with multiplication. Thus talking about surjective group homomorphisms out of (or quotient groups of ), up to a suitable notion of isomorphism, is equivalent to talking about congruence relations on , which is in turn equivalent to talking about normal subgroups of .
Equivalence relations and quotients of rings
Let be a ring and let be a ring homomorphism. As before, the relation is an equivalence relation on . But since it must also respect addition and multiplication, satisfies
- compatibility with addition: if and then ,
- compatibility with multiplication: if and then .
This defines a congruence relation on rings. Since addition has inverses, we conclude that is equivalence modulo for some normal subgroup of (under addition), namely the kernel of as a homomorphism of additive groups. Since addition is commutative, we can drop the adjective “normal.” Note that compatibility with addition and multiplication, in the presence of the other axioms, is equivalent to the condition that the equivalence relation, as a subset
of , is a subring.
As before, we get a quotient map of rings. In addition, compatibility with multiplication implies that
Thus is closed under left and right multiplication by elements of : it is a two-sided ideal. Conversely, if is a two-sided ideal, then the corresponding equivalence relation has the following property: if , then
from which it follows that
hence . So being a two-sided ideal is equivalent to being compatible with multiplication. Thus talking about surjective ring homomorphisms out of (or quotient rings of ) is equivalent to talking about congruence relations on , which is in turn equivalent to talking about two-sided ideals of .
Equivalence relations and quotients for monoids
When we try to repeat the above discussion in the setting of monoids we run into the problem that a congruence on a monoid is not completely determined by what is equivalent to the identity since we can no longer rely on inverses.
Example. Consider the monoids . The monoid admits a surjective homomorphism to sending a generator to a generator. It is not injective, but the kernel is trivial.
Thus monoids resemble more closely the situation for sets and topological spaces: we have to talk about congruence relations (namely submonoids of which are also equivalence relations) and talking about these isn’t equivalent to talking about special kinds of submonoids of .
Equivalence relations and quotients for topological spaces
We close with a less algebraic example. Let be a topological space and let be a continuous function to another topological space . As before, the relation is an equivalence relation on . In the setting of general topological spaces, we cannot say any more about since it can in fact be arbitrary: the set of equivalence classes with respect to any equivalence relation may be given the quotient topology, which is by definition the universal topology such that the quotient map is continuous.
However, if is Hausdorff (in particular if we are working in a subcategory of the category of Hausdorff spaces), then the equivalence relation, as a subset
of , is the preimage of the diagonal under the map
Since is Hausdorff, the diagonal is closed (this is equivalent to being Hausdorff!), so its preimage is also closed. Thus the equivalence relation itself must be a closed subspace of . If we restrict ourselves to the category of compact Hausdorff spaces, then it follows that the equivalence relation is a compact Hausdorff subspace of .