Let . Then becomes a group under coset multiplication.
Define the * quotient map* (or *
canonical projection*) by

* Proposition.* If , the quotient map is a surjective
homomorphism with kernel H.

* Proof.* If , then

Therefore, is a group map.

Obviously, if , then . Hence, is surjective.

Finally, I'll show that . If , then , and H is the identity in . Therefore, , so .

Conversely, suppose . Then , so , so . Therefore, , and hence .

The preceding lemma shows that *every normal subgroup is the
kernel of a homomorphism*: If H is a normal subgroup of G, then
, where is the quotient map. On the other
hand, the kernel of a homomorphism is a normal subgroup.

* Corollary.* Normal subgroups *are
exactly* the kernels of group homomorphisms.

Normality was defined with the idea of imposing a condition on subgroups which would make the set of cosets into a group. Now an apparently independent notion --- that of a homomorphism --- gives rise to the same idea! This strongly suggests that the definition of a normal subgroup was a good one.

You can think of quotient groups in an even more subtle way. The
general theme is something like this. In modern mathematics, it is
important to study not only objects --- like groups --- but the maps
between objects --- in this case, group homomorphisms. The maps,
after all, describe the *relationships* between different
objects. (This theme is elaborated in a branch of mathematics called
* category theory*.)

It turns out that more is true. In a sense, the maps carry *all of
the information* about the objects; one could even be perverse
and "build up" objects out of maps! I won't go to such
extremes, but in some cases, an object can be *characterized*
by certain maps. Here's an important example.

* Theorem.* (* Universal Property
of the Quotient*) Let , and let be a group
homomorphism such that . Then there is a unique
homomorphism such that the following diagram commutes:

(To say that the diagram * commutes* means that
.)

* Proof.* Define by

This is forced by the requirement that , since plugging into both sides yields , or .

I need to check that this map is * well-defined*.
The point is that a given coset may in general be written as , where . I must verify that the result
or is the same regardless of how I
write the coset.

(If in this situation, then a single input --- the coset --- produces different outputs, which contradicts what it means to be a function.)

So suppose that , so for some .

This shows that is indeed well-defined.

I was forced to define as I did in order to make the diagram commute. Hence, is unique.

Now I'll show that is a homomorphism. Let . Then

Therefore, is a homomorphism.

The universal property of the quotient is an important tool in constructing group maps: To define a map out of a quotient group , define a map out of G which maps H to 1.

The map *you* construct goes from G to ; the universal property
*automatically* constructs a map for you. The advantage of using
the universal property rather than defining a map out of directly is that you don't repeat the
verification that the map is well-defined --- it's been done once and
for all in the proof above.

Should you ever need to know how the magic map is defined, refer to the proof (and the commutativity of the diagram).

* Remarks.* (a) Many other constructions are
characterized by universal properties. In each case, one finds that
the appropriate conditions imply the existence of a unique map with
certain properties.

(a) The use of diagrams of maps --- particularly commutative ones ---
is pervasive in modern mathematics. They are a powerful language, and
another outgrowth of the categorical point of view. In general, one
says a diagram * commutes* if following the
"paths" indicated by the arrows (maps) in different ways
between two objects produces the same result. For example, consider
the diagram

To say that this diagram commutes means that .

* Example.* Use the universal property to show
that
given by is a
well-defined group map.

I can regard as . To define f, begin by defining by

Let . Then since 24 is a multiple of 12,

This means that maps the subgroup of to the identity . By the universal property of the quotient, induces a map given by

I can identify with by reducing mod 8 if needed. (Thus, is identified with .) Then the definition of f becomes

This is the group map I wanted to construct.

* Example.* (* Using the
universal property to construct a group map*) Use the universal
property to construct a homomorphism from the quotient group to .

The universal property tells me to construct a group map from to which contains in its kernel --- that is, which sends to 0. Now consists of all multiples of , so what I'm looking for is a group map which sends to 0.

To ensure that what I get is a *group map*, I should probably
guess a linear function --- something like

If , then . There is no question of
*solving* this equation for a and b, since there is one
equation and two variables. But I just need *some* a and b
that work --- and one "obvious" way to do this is to set
and , since

Notice that , would work, too. In fact, there are infinitely many possibilities.

So I define by

It's easy to check that this is a group map, and I constructed it so
that . Therefore, the universal property
*automatically* produces a group map . It is defined by

Why not just define the map this way to begin with? If you did, you'd
have to check that the map was *well-defined*. It's less messy
to use the universal property to construct the map as above.

Copyright 2018 by Bruce Ikenaga