The basic idea

Let’s say we have a group. Maybe the group is kinda large and unwieldy, and we want to find an easier way to think about it. Or maybe we just want to focus on a certain aspect of the group. Some of the actions will change things in ways we just don’t really care about, or don’t mind ignoring for now. So let’s create a group homomorphism that will map all these actions to the identity action in a new group. The image of this homomorphism will be a group much like the first, except that it will ignore all the effects that come from those actions that we’re ignoring—just what we wanted! This new group is called the quotient group.

Definition

We start with our group \(G\). The actions we want to ignore form a group \(N\), which must be a normal subgroup of \(G\). The quotient group is then called \(G/N\), and has a canonical homomorphism \(\phi: G \rightarrow G/N\) which maps \(g \in G\) to the coset \(gN\).

The divisor group

In the definition, we require the divisor \(N\), to be a normal subgroup of \(G\). Why? Well first, let’s see why requiring \(N\) to be a group makes sense. Remember that \(N\) has the actions whose effects we want to ignore. So it makes sense that it should contain the identity action, which has no effect. It also is reasonable that it would be closed under the group operation—doing two things we don’t care about shouldn’t change anything we care about. Together, these two properties imply it is a subgroup: \(N \le G\).

A subgroup is great, but it isn’t quite good enough by itself to work here. That’s because we want the quotient group to preserve the overall structure of the group, i.e. it should preserve the group multiplication. In other words, there needs to be a group homomorphism \(\phi\) from \(G\) to \(G/N\). Since \(N\) is the subgroup of things we want to ignore, all its actions should get mapped to the identity action under this homomorphism. That means it’s the kernel of the homomorphism \(\phi\), which means it’s a normal subgroup: \(N \trianglelefteq G\).

Cosets

What exactly are the elements of the new group? They are equivalence classes of actions, the sets \(gN = \{gn : n \in N\}\) where \(g \in G\), also known as a coset. The identity element is the set \(N\) itself. Multiplication is defined by \(g_1N \cdot g_2N = (g_1g_2)N\).

Generalizes the idea of a quotient

What gives a quotient group the right to call itself a quotient? If \(G\) and \(N\) both have finite order, then \(|G/N| = |G|/|N|\), which can be proved by the fact that \(G/N\) consists of the cosets of \(N\) in \(G\), and that these cosets are the same size, and partition \(G\).

Example

Suppose you have a collection of objects, and you need to split them into two equal groups. So you are trying to determine under what circumstances changing the number of objects will affect this property. You notice that changing the size of the collection by certain numbers such as 0, 2, 4, 24, and −6 doesn’t affect this property.

The set of different size changes can be modeled as the additive group of integers \(\mathbb Z\). The changes that don’t affect this property also form a group: \(2\mathbb Z = \{2n : n\in \mathbb Z\}\). Exercise: verify that this is a normal subgroup of \(\mathbb Z\).

This subgroup gives us two cosets: \(0 + 2\mathbb Z\) and \(1 + 2\mathbb Z\) (remember that \(+\) is the group operation in this example), which are the elements of our quotient group. We will give them their conventional names: \(\text{even}\) and \(\text{odd}\), and we can apply the coset multiplication rule to see that \(\text{even}+ \text{even} = \text{even}\), \(\text{even} + \text{odd} = \text{odd}\), and \(\text{odd} + \text{odd} = \text{even}\).

Instead of thinking about specific numbers, and how they will change our ability to split our collection of objects into two equal groups, we now have reduced the problem to its essence. Only the parity matters, and it follows the simple rules of the quotient group we discovered.

Quotient group