Quotient group

A quotient group is a special kind of mathematical group made from a larger group by grouping together similar elements. Think of it like sorting a big pile of toys into smaller, neat groups based on certain rules.

One common example is the cyclic group of addition modulo n. This can be created from the group of all integers by grouping numbers that differ by a multiple of n. Each group of numbers that are the same in this way is called a congruence class. By treating each class as a single item, we get a new, smaller group.

Quotient groups are very important in group theory because they are closely connected to homomorphisms. A homomorphism is a special kind of map between groups. The first isomorphism theorem tells us that the result of applying a homomorphism to a group is always isomorphic to a quotient group of the original group. This helps mathematicians understand the structure of groups better.

Definition and illustration

A quotient group is a way to make a new group from an old one by putting together similar parts. Imagine you have a big group and you want to make it simpler by grouping things that act the same way. This helps us understand how groups are built better.

For example, think of numbers you can add, like 0, 1, 2, 3, 4, and 5, where adding makes you start again at 0 after 5. If we group these numbers into pairs that are alike, such as pairing 0 with 3 and 1 with 4, we can make a smaller group. This smaller group still follows the same addition rules. We call this smaller group a quotient group, and it helps us see patterns in the big group more clearly.

Motivation for the name "quotient"

The idea behind a quotient group is like dividing numbers. For example, when you divide 12 by 3, you get 4 because you can split 12 objects into 4 groups of 3 objects each. A quotient group works in a similar way, but instead of getting a number, you get another group.

In a quotient group, we start with a big group and a smaller special group called a normal subgroup. By grouping parts of the big group in a certain way, we end up with a new, smaller group that keeps some of the original group's properties. This new group is called the quotient group.

Examples

Even and odd integers

Imagine you have a group of all whole numbers, like ..., -2, -1, 0, 1, 2, .... When we add these numbers together, we get another number in the same group. Now, let’s look at just the even numbers: ..., -4, -2, 0, 2, 4, .... This smaller group fits inside the bigger group.

We can split the big group into two parts: the even numbers and the odd numbers. Each part acts like a single item. When we combine these parts using addition, we get a new small group with just two items. This small group is like a clock that only counts to 2.

Remainders of integer division

Let’s take our group of whole numbers again. Pick any whole number, like 5. If we only care about remainders after dividing by 5, we can group numbers by their remainders: 0, 1, 2, 3, or 4. This creates a new group where we only work with these remainders.

Complex integer roots of 1

The twelfth roots of unity are points on a circle in the complex plane. They form a group under multiplication. A subgroup of these points, the fourth roots of unity, also forms a group. By looking at how these subgroups relate, we can split the bigger group into three parts, each acting like a single color.

Real numbers modulo the integers

Consider all real numbers, like 3.14 or -2.71. We can group these numbers by how much they differ from whole numbers. Each group looks like a slice of the number line between two whole numbers. When we add these slices together, we get a new group that acts like a circle.

Matrices of real numbers

We can look at groups made of special tables of numbers called matrices. Some of these matrices have a special property: their size determinant is 1. These form a subgroup. By comparing all matrices to this subgroup, we can split them into groups based on their determinants.

Integer modular arithmetic

Take a small group of numbers like 0, 1, 2, 3, where we add only up to 3 (so 1 + 3 = 0). Inside this group, we can find a smaller group like {0, 2}. By looking at how these smaller groups fit inside the bigger one, we can create a new group with just two items.

Integer multiplication

In number theory, we study groups formed by numbers under multiplication. For a special number n, we can look at numbers that stay the same when multiplied by n. These form a subgroup. By comparing all numbers to this subgroup, we can split them into groups that help with certain types of secret codes.

Properties

The quotient group ( G/N ) has special features based on the group ( G ) and the subgroup ( N ). For example, if ( G ) has only one element, then ( G/N ) is also very simple. If ( N ) is the identity element, then ( G/N ) is the same as ( G ).

The size of ( G/N ) depends on how ( N ) fits inside ( G ). Even if both ( G ) and ( N ) are large, ( G/N ) can be small or even have no end to its size. There is a natural way to match elements of ( G ) to ( G/N ), which helps us see how these groups connect.

Quotients of Lie groups

If you have a special kind of mathematical group called a Lie group, and you take a smaller part of it that also has special properties, you can create a new group by grouping together similar elements. This new group is called a quotient group.

In this case, the bigger group has a structure like a bundle of fibers, with the new quotient group as the base and the smaller part as the fiber. The size of this new group depends on the sizes of the original group and the smaller part.