Free group

In mathematics, a free group is a special kind of group made from a set of elements called generators. These generators can be combined in many ways to form what are called "words." Two words are considered the same only if they can be shown to be equal using the basic rules that define groups. This means that each element in the free group can be written in exactly one way using the generators and their opposites, or inverses.

The number of generators in a free group is known as its rank. A group is called free if it matches the structure of a free group for some subset of its elements. Free groups are important because they help mathematicians understand the basic building blocks of more complex groups. They are a specific example of a free object in the study of universal algebra.

History

Free groups were first discovered while studying hyperbolic geometry, especially as examples of Fuchsian groups. In 1882, Walther von Dyck showed that these groups have very simple descriptions. Later, in 1924, Jakob Nielsen began studying them algebraically and gave them their name. Max Dehn linked free groups to topology and proved an important theorem, with Otto Schreier later providing an algebraic proof in 1927. Kurt Reidemeister also wrote about free groups in his 1932 book on combinatorial topology.

Examples

The group of integers is a free group with just one generator, which is the number 1. This group is special because it is also a free abelian group. However, free groups with two or more generators are not abelian.

Free groups with two generators appear in some interesting math proofs, like the one for the Banach–Tarski paradox.

In algebraic topology, the fundamental group of a shape made of k loops joined at one point is a free group with k generators.

Construction

The free group (F_S) with free generating set (S) can be built in a special way. Think of (S) as a set of symbols. For every symbol (s) in (S), we also have an “inverse” symbol, written (s^{-1}).

A word in (S) is any combination of these symbols and their inverses. For example, if (S = {a, b, c}), then we can make words like (ab^3c^{-1}ca^{-1}c).

We can simplify words by removing pairs like (c) and (c^{-1}) when they appear next to each other. After simplifying as much as possible, we get a reduced word. The free group (F_S) is the collection of all reduced words, where we combine words by writing them one after another and then simplifying if needed. The empty word (with no symbols) acts as the identity element.

A reduced word is cyclically reduced if the first and last symbols are not inverses of each other. Every word can be changed into a cyclically reduced word by rearranging, and doing this gives us different ways to look at the same word. For example, (b^{-1}abc b) is not cyclically reduced, but it can be rearranged to (abc), which is.

Universal property

The free group is a special kind of group made from a set of symbols. It follows a rule called the "universal property." This means that for any group you choose and any way to connect the symbols to elements of that group, there is exactly one way to create a matching rule for the whole free group.

In simple terms, the free group lets you combine symbols in many ways, but it follows the basic rules of groups. This makes it a building block for understanding more complex groups. The set of symbols you start with is called a "basis" for the free group.

Facts and theorems

Free groups have some interesting properties that come from how they are defined:

1. Any group can be made from a free group by choosing some generators. If the set of generators is finite, the group is called finitely generated.
2. If the set of generators has more than one element, the free group is not abelian, meaning the order of operations matters.
3. Two free groups are the same if and only if their sets of generators have the same number of elements. This number is called the rank of the free group.

Some other important facts include:

1. Every subgroup of a free group is also free.
2. A free group with at least two generators has subgroups of many different sizes.
3. The Cayley graph of a free group looks like a tree, which helps us understand its structure.

Free abelian group

Further information: Free abelian group

A free abelian group on a set is a special kind of group made from that set. It follows certain rules that make it easier to work with, similar to how we add and subtract numbers.

Basically, the free abelian group on a set can be thought of as all the possible ways to combine the elements of that set, but where the order doesn’t matter. This means that putting elements together in different orders gives the same result. The number of elements in the original set is called the rank of the free abelian group.

Tarski's problems

Around 1945, a mathematician named Alfred Tarski asked some important questions about free groups. He wanted to know if free groups with two or more generators share the same basic rules and if these rules can be figured out step by step. Later work showed that any two free groups that are not simple have the same basic rules, and these rules can indeed be solved.

There is still an open question in a related area of mathematics: whether certain representations of free groups are the same for any two free groups with a limited number of generators.