Free group

by Patrick Stevens Jul 23 2016 updated Oct 23 2016

The free group is "the purest way to make a group containing a given set".

[summary: The free Group $~$F(X)$~$ on the set $~$X$~$ is the group whose elements are the freely reduced words over $~$X$~$, and whose group operation is "[-concatenation] followed by free reduction".]

Intuitively, the free Group $~$F(X)$~$ on the set $~$X$~$ is the group whose elements are the freely reduced words over $~$X$~$, and whose group operation is "[-concatenation] followed by free reduction".

The free group can be constructed rigorously in several equivalent ways, some of which are easy to construct but hard to understand, and some of which are intuitive but rather hard to define properly. Our formal construction (detailed on the Formal Definition lens) will be one of the more opaque definitions; there, we eventually show that the formal construction yields a group which is isomorphic to the intuitive version, and this will prove that the intuitive version does in fact define a group with the right properties for the free group.

Intuitive definition

Given a set $~$X$~$, the free group $~$F(X)$~$ (or $~$FX$~$) on $~$X$~$ has:


The free group on the set $~$X = \{ a, b \}$~$ contains the following elements (among others):

(From now on, we will only use the common shorthand to denote words, except in cases where this interferes with something else we're doing.)

Some things which the same free group does not contain are:

Some examples of this group's group operation (which we will write as $~$\cdot$~$) in action are:


Why are the free groups important?

It is a fact that Every group is a quotient of a free group. Therefore the free groups can be considered as a kind of collection of "base groups" from which all other groups can be made as quotients. This idea is made more concrete by the idea of the Group presentation, which is a notation that specifies a group as a quotient of a free group. %%note:Although it's not usually presented in this way at first, because the notation has a fairly intuitive meaning on its own.%%

Free groups "have no more relations than they are forced to have"

The crux of the idea of the group presentation is that the free group is the group we get when we take all the elements of the set $~$X$~$ as elements which "generate" our putative group, and then throw in every possible combination of those "generators" so as to complete it into a bona fide group. We explain why the free group is (informally) a "pure" way of doing this, by walking through an example.

%%%hidden(Example): If we want a group which contains the elements of $~$X = \{ a, b \}$~$, then what we could do is make it into the Cyclic group on the two elements, $~$C_2$~$, by insisting that $~$a$~$ be the identity of the group and that $~$b \cdot b = a$~$. However, this is a very ad-hoc, "non-pure" way of making a group out of $~$X$~$. %%note:For those who are looking out for that sort of thing, to do this in generality will require heavy use of the axiom of choice.%% It adds the "relation" $~$b^2 = a$~$ which wasn't there before.

Instead, we might make the free group $~$FX$~$ by taking the two elements $~$a, b$~$, and throwing in everything we are forced to make if no non-trivial combination is the identity. For example:

This way adds an awfully large number of elements, but it doesn't require us to impose any arbitrary relations. %%%

Universal property of the free group

The free group has a universal property, letting us view groups from the viewpoint of Category theory. Together with the idea of the quotient (which can be formulated in category theory as the [-coequaliser]) and the subsequent idea of the Group presentation, this lets us construct any group in a category-theoretic way.

Indeed, every group $~$G$~$ has a presentation $~$\langle X \mid R \rangle$~$ say, which expresses $~$G$~$ as a quotient (i.e. a certain coequaliser) of the free group $~$F(X)$~$. We can construct $~$F(X)$~$ through the universal property, so we no longer need to say anything at all about the elements or group operation of $~$G$~$ to define it.


%%hidden(Proof that "the rationals under addition" is torsion-free but not free): Suppose the order of an element $~$x \in \mathbb{Q}$~$ were finite and equal to $~$n \not = 0$~$, say. Then $~$x+x+\dots+x$~$, $~$n$~$ times, would yield $~$0$~$ (the identity of $~$(\mathbb{Q}, +)$~$). But that would mean $~$n \times x = 0$~$, and so $~$n=0$~$ or $~$x = 0$~$; since $~$n \not = 0$~$ already, we must have $~$x = 0$~$, so $~$x$~$ is the identity after all.

The group is not free: suppose it were free. It is abelian, so it must be isomorphic to either the trivial group (clearly not: $~$\mathbb{Q}$~$ is infinite but the trivial group isn't) or $~$\mathbb{Z}$~$. It's not isomorphic to $~$\mathbb{Z}$~$, though, because $~$\mathbb{Z}$~$ is cyclic: there is a "generating" element $~$1$~$ such that every element of $~$\mathbb{Z}$~$ can be made by adding together (possibly negatively-many) copies of $~$1$~$. $~$\mathbb{Q}$~$ doesn't have this property, because if $~$x$~$ were our attempt at such a generating element, then $~$\frac{x}{2}$~$ could not be made, so $~$x$~$ couldn't actually be generating after all. %%


Eric Rogstad

The free group on the set $~$X \= \\{ a, b \\}$~$ contains some of the following elements:

This wording suggests the group contains only some of the elements from the following list. I think you meant the opposite -- that the following elements are only some of the things it contains.

Jaime Sevilla Molina

Then we do it all again for all the inverses $~$x^{-1}$~$, creating the functions $~$\\rho\_{x^{-1}}$~$\.

Pedantic remark: Aren't you missing the identity of free groups in your intuitive construction?

We have the $~$\rho_x$~$ and the $~$\rho_{x^{-1}}$~$. Where is $~$\rho_\epsilon$~$?