Understanding the quotient map

The notions of normal subgroup, quotient map and quotient group are somewhat tricky to understand the first time. These notions are also extremely crucial to the structure theory of groups. In this survey article, we study the differing approaches we can take to studying and understanding the quotient map. We begin with a few examples.

Quotient maps for sets
In arithmetic, we think of a quotient as a division of one number by another. While this description is somewhat relevant, it is not the most appropriate for quotient maps of groups. A better way is to first understand quotient maps of sets.

A quotient map, and surjective homomorphism, of sets
Suppose $$A$$ and $$B$$ are two sets, and $$f:A \to B$$ is a surjective function (or set map). In other words, for every $$b \in B$$, the set $$f^{-1}(b) = \{ a \in A \mid f(a) = b \}$$ is nonempty. Then, we can associate, to each $$b \in B$$, the set $$f^{-1}(b)$$.

Observe that:


 * The sets $$f^{-1}(b)$$ form a partition of $$A$$ into disjoint, nonempty subsets.
 * Two elements $$a_1, a_2 \in A$$ are in the same part if and only if $$f(a_1) = f(a_2)$$
 * So, the sets $$f^{-1}(b)$$ are the equivalence classes in $$A$$ under the equivalence relation $$a_1 \sim a_2 \iff f(a_1) = f(a_2)$$

Now, suppose that instead of a map $$f: A \to B$$, we started with an equivalence relation $$\sim$$ on $$A$$. Then, we could construct a set $$A/(\sim)$$ as the set of equivalence classes under $$\sim$$, and define a map $$f':A \to A/\sim$$ as sending each $$a \in A$$ to its equivalence class. This is termed the quotient map for the equivalence relation $$\sim$$.

Thus, we have:


 * A way of using a surjective function to construct an equivalence relation (namely, the equivalence relation of having the same image)
 * A way of using an equivalence relation to generate a surjective function (namely, the quotient map of that equivalence relation)

These two associations are inverses of each other in a weak sense. Namely, if we start with a function $$f:A \to B$$, obtain the equivalence relation, and then consider the quotient map $$f':A \to A/\sim$$, then we can construct a natural bijection $$g: A/\sim \to B$$ such that $$f = g \circ f'$$. The bijection simply sends an equivalence class of elements in $$A$$ to the image of any of those elements, in $$B$$.

Conversely, if we start with an equivalence relation $$\sim$$, construct the quotient map, and take the equivalence relation arising from that quotient map, we recover the same equivalence relation $$\sim$$.

Here's an example. Suppose $$A = \{ 1,2,3 \}$$, $$B = \{ 4,5 \}$$ and $$f$$ is given by:

$$f(1) = f(3) = 4, f(2) = 5$$

Then the equivalence relation generated has equivalence classes $$\{ 1,3\}$$ and $$\{ 2 \}$$. The quotient is the set:

$$\{ \{1,3 \}, \{ 2 \} \}$$

with the map $$f'$$ being:

$$f'(1) = f'(3) = \{ 1, 3 \}, f'(2) = \{ 2 \}$$

and the natural bijection $$g$$ is given by:

$$g(\{ 1,3 \}) = 4, g( \{ 2 \}) = 5$$

Quotient maps and congruences
We'd like to say something analogous to what was said for sets, but respecting the group structure. In other words, we want to relate surjective homomorphisms of groups, with quotient maps from certain equivalence relations.

One direction is easily done. Given any surjective homomorphism $$\varphi:G \to H$$, we obtain an equivalence relation on $$G$$. However, not every equivalence relation on $$G$$ can arise from a surjective homomorphism. The equivalence relation has some special properties. Let's look at this more closely.

The homomorphism condition tells us that if $$a_1,a_2 \in G$$, then $$\varphi(a_1a_2) = \varphi(a_1)\varphi(a_2)$$. In particular, it tells us that the image of $$a_1a_2$$ depends only on the images of $$a_1$$ and $$a_2$$. Thus, we have:

$$\varphi(a_1) = \varphi(b_1), \varphi(a_2) = \varphi(b_2) \implies \varphi(a_1a_2) = \varphi(b_1b_2)$$

This imposes the following condition on the equivalence relation:

$$a_1 \sim b_1, a_2 \sim b_2 \implies a_1a_2 \sim b_1b_2$$

Similarly, we obtain that:

$$a \sim b \implies a^{-1} \sim b^{-1}$$

An equivalence relation $$\sim$$ satisfying the above conditions is termed a congruence on a group. We now see that, starting with any congruence on a group, we can take the corresponding quotient map, and give a group structure to the quotient, so that the map is a surjective homomorphism. Namely, given two equivalence classes, we multiply their representatives, and take the equivalence class of the product.

So, we have two associations:


 * A surjective homomorphism gives rise to a congruence, namely, the equivalence relation of having the same image.
 * A congruence gives rise to a surjective homomorphism, namely, its quotient map.

Further, the two associations are inverses of each other, in the following weak sense. Suppose $$\varphi:G \to H$$ is a surjective homomorphism of groups, and $$\sim$$ is the equivalence relation that this generates on $$G$$. This equivalence relation is a congruence. Consider the quotient map $$\varphi': G \to G/\sim$$. Then, there is a natural isomorphism $$\sigma:G/\sim \to H$$ such that $$\sigma \circ \varphi' = \varphi$$.

A quick preview
On a set, we can have practically any kind of equivalence relation. The equivalence classes may have different sizes, and knowing one equivalence class may not determine what the other equivalence classes look like.

A congruence on a group, however, is extremely controlled, in three important ways:


 * All the equivalence classes have the same size.
 * The equivalence class containing the identity element is a normal subgroup, and every normal subgroup can be realized as the equivalence class of the identity element under a congruence. This equivalence class is also the kernel of the homomorphism to the quotient group.
 * The equivalence class of the identity element determines the entire congruence (the congruence classes are the cosets of the normal subgroup). In particular, it determines the quotient map. Thus, given any normal subgroup, we can talk of the quotient map corresponding to that normal subgroup. This allows us to reformulate the correspondence between surjective homomorphisms and congruences, as a correspondence between surjective homomorphisms and normal subgroups. That's how the first isomorphism theorem is usually stated.

This is in sharp contrast with sets, and somehow shows that the structure of groups is more uniform and better controlled by what happens at some places.

Viewing this in terms of the variety of groups
One approach to studying algebraic structures is universal algebra, where we look at sets with arbitrary collections of operations satisfying identities. In this language, the variety of groups is given by three operations (multiplication, inverse and the identity element) satisfying the identities of associativity, identity element, and inverses. Here, an individual group is treated as an algebra in the variety of groups.

We can define a congruence on an algebra as an equivalence relation that is respected by all the algebra operations. This gives rise to the usual notion of congruence on a group: an equivalence relation on the group that is preserved by the multiplication and the inverse map. Then, it is true for a general variety of algebras that:


 * Any surjective homomorphism from an algebra gives rise to a congruence on that algebra
 * Any congruence on an algebra gives rise to a surjective homomorphism from that algebra, namely, the quotient map
 * These two maps are inverses of each other in the weak sense (as described for groups and sets)

What makes the variety of groups different is that the congruences are particularly nice. In the language of universal algebra, this says:


 * 1) Variety of groups is ideal-determined: A congruence on a group is completely determined by the equivalence class of the identity element, which is a normal subgroup, and any normal subgroup comes from a congruence.
 * 2) Variety of groups is congruence-uniform: The equivalence classes under any congruence on a group are equal in size.

There are other interesting facts about the quotient map for groups that makes them better-behaved than other algebraic structures, such as the fact that ideals are subalgebras in the variety of groups (every normal subgroup is a subgroup) and characteristic subalgebras are ideals in the variety of groups (characteristic subgroups are normal).

The trivial congruence
The trivial congruence is the congruence where any two elements of the group are congruent. In this case, there is only one congruence class.


 * The quotient group is the trivial group, and the quotient map is the map sending all elements to the identity element of the trivial group.
 * The kernel is the whole group, which is clearly a normal subgroup of itself.

The trivial congruence is the coarsest congruence: it has the least ability to distinguish elements of the group. In fact, it cannot distinguish elements at all.

The discrete congruence
The discrete congruence is the congruence where two elements are in the same congruence class if and only if they are equal. In this case, the congruence classes are singleton sets.


 * The quotient group is the group itself, and the quotient map is the identity map.
 * The kernel is the trivial subgroup.

The discrete congruence is the finest congruence on a group: it has the most ability to distinguish elements of the group. In fact, it can distinguish any two different elements of the group.

These are the extreme cases
The discrete and trivial congruence are extreme cases, and all other congruences lie in between: the congruence classes have sizes more than one, but they are all proper subsets of the group. Quotients arising this way are termed proper nontrivial quotients, and their kernels are proper nontrivial normal subgroups: normal subgroups that are neither trivial nor equal to the whole group.

A group for which there are no other congruences is termed a simple group.

Modular arithmetic
Consider the group of integers under addition, that we'll denote $$\mathbb{Z}$$. This is an Abelian group, with identity element zero. For any nonzero integer $$n$$, we can define an equivalence relation by:

$$a \sim b \iff n | a - b$$

In other words, two integers are equivalent iff their difference is a multiple of $$n$$. This equivalence relation is a congruence in the sense that:

$$a \sim b, c \sim d \implies a + c \sim b + d$$

and:

$$a \sim b \implies -a \sim -b$$

Thus, we can take the quotient of the group of integers by this equivalence relation, to get a group.

Let's first look at the equivalence classes. The equivalence class of zero under this equivalence relation is the set of all multiples of $$n$$. The equivalence class of 1 is the set of integers of the form $$1 + kn$$ where $$k$$ is an integer. This is the same as those elements that leave a remainder of 1 on division by $$n$$. More generally, the equivalence class of $$a$$ is the set of all elements of the form $$a + kn$$. This is termed the congruence class of $$a$$.

There is a total of $$n$$ congruence classes, and each congruence class has a unique representative among the integers $$0,1,2,\dots, n-1$$. This unique representative equals the remainder on dividing any element of the congruence class, by $$n$$.

Now, we observe that the equivalence class of the identity element, namely, the set of multiples of $$n$$, is a normal subgroup. The quotient group we get is a group of size $$n$$, where, instead of adding elements, we add their congruence classes.

For instance, suppose $$n = 5$$. Then, the quotient group is a group of order five. Call the elements of this group $$0,1,2,3,4$$, where each element stands for the congruence class it represents in $$\mathbb{Z}$$. Then, $$1 + 2 = 3$$, while $$3 + 4 = 2$$, because in the group of integers, $$3 + 4 = 7$$, which is congruent to $$2$$ mod $$5$$.

Let's use modular arithmetic to understand how simply knowing a normal subgroup determines the congruence. Suppose we're given, in the group $$\mathbb{Z}$$, the normal subgroup $$n\mathbb{Z}$$. We need to find a congruence $$\sim$$ with this as the congruence class of zero. Note first that a congruence is an equivalence relation, so for any $$a \in \mathbb{Z}$$, we have, for any $$k \in \mathbb{Z}$$:

$$a \sim a \ \forall \ a \in \mathbb{Z}, 0 \sim kn \ \forall \ k \in \mathbb{Z}$$

Since $$\sim$$ is a congruence, we are forced to conclude that:

$$a \sim a + kn \ \forall \ a,k \in \mathbb{Z}$$

Thus, if two elements differ by a multiple of $$n$$, they are in the same congruence class.

Conversely, suppose $$a \sim b$$. Then, since $$-b \sim -b$$, we get that:

$$a + (-b) \sim b + (-b) = 0$$

So $$a - b \in n \mathbb{Z}$$, and thus, any two elements in the same congruence class must differ by a multiple of $$n$$.

Thus, two elements are in the same congruence class iff they differ by a multiple of $$n$$. This illustrates how the normal subgroup completely determines the congruence, and hence, the quotient map.

Vector spaces
Consider $$\R^2$$ as a vector space over $$\R$$. Consider the subspace $$A = \{ (x,y) \mid x = y \}$$. This is a subgroup in the additive sense, and it is normal (because the whole group is Abelian). We need to find a congruence with this normal subgroup as the kernel.

Consider any two elements $$(x_0,y_0), (x_1,y_1) \in \R^2$$. We know that these two elements are in the same congruence class if and only if their difference is in the subspace $$A$$. This means that $$x_0 - x_1 = y_0 - y_1$$, which gets rewritten as $$x_0 - y_0 = x_1 - y_1$$. Thus, the congrience classes are parametrized by the difference of the coordinates.

This can be seen geometrically. The subspace $$A$$ in $$\R^2$$ is the line $$x = y$$, and its cosets are all the lines parallel to the line $$x = y$$, namely, lines of the form $$x - y = c$$ for some $$c \in \R$$.

The quotient group is isomorphic to $$\R$$. When we add two cosets, we add the corresponding $$c$$ values. Thus, the sum of the line $$x - y = 1$$ and the line $$x - y = 2$$ is the line $$x - y = 3$$.

Examples in non-Abelian groups
Non-Abelian groups are a bit trickier because not every subgroup is normal. Nonetheless, we can study some interesting examples.