# The free group

A free object (say a group, an algebra or a topological space) over a set can be thought of as the generic object of that type containing a copy of In other words, the only relations between the elements of the free object are the relations needed for it to satisfy it’s defining properties. Even if you never heard about the notion of a free object explicitly, the idea should be familiar. For example, the free vector space over a set is simply the vector space with basis After constructing the free group, speaking of “the” free object over will be shown to be justified. Indeed, as a consequence of the fact that free objects satisfy a so-called universal property, which could thus be used as a definition, they are unique up to isomorphism.

As an illustrative example of a free object, we’ll construct the free group over a set The idea will be to consider the set of all words in a well-chosen alphabet together with the operation of concatenation of words, and then to impose the minimal relations between these words in order for this set to be a group. Take a set in bijection with but disjoint from it. For convenience, for each let’s note it’s corresponding element and vice versa (so that ). Our last ingredient will be a singleton with and Note the union of these 3. In other words, is the disjoint union of with itself and a singleton. Now denote the set of all finite words on the alphabet i.e.

Define the concatenation of words on the alphabet by

where Clearly is closed under concatenation of words on It only remains to give it a group structure by introducing some relations between some of its elements. But the key idea in the notion of a free object, is to introduce only the minimal relations. In this case, we want an identity element and an inverse for every element. This will be achieved minimally with the following relation :

The resulting set of equivalence classes with the obvious operation induced by concatenation of words, noted is what is called a free group over and is often denoted by It would now only remain to verify that is indeed a group, which I will not do.

Note that the set can be viewed as being contained in by associating the 1-letter word to every This inclusion map will be denoted by

A very important property of free groups is that they satisfy the following universal property:

Theorem: (universal property)Let be a group, a set and a map from to Then there exists a unique group homomorphism such that for all i.e. this diagram commutes :

**Proof:** Let be defined on with for all Then, since generates there is a unique way of extending to a homomorphism defined on all of Namely, we need to have

where and is identified with . Note that this is the same idea as in linear algebra where a linear map is entirely determined by it’s action on a set of generators for the vector space.

QED

So, why is this property so important? Well, it guarantees that the free group over is (up to isomorphism) the most general group containing In a sense, it guarantees us that if what we want is a master group containing all the information about the set (so that we lose nothing in passing from to our generic group), this is exactly what we got. This property could even be used as the definition of a free group :

Corollary:All groups satisfying the universal property of free groups (satisfying the theorem) are isomorphic.

**Proof:** Suppose and are two groups satisfying the universal property, in other words, they are two free groups generated by Denote by (resp ) the natural injection from to (resp ) and identify with its image by in . Then by the universal property, there exists unique group homomorphisms and such that **Id**. Since the composition is in **End** and is the identity on by uniqueness it must be the identity map. Similarly, **Id** thus is an isomorphism.

QED

Finally, for any group consider the identity homomorphism of as a set map from to Then there exists a unique such that so is surjective. By the first isomorphism theorem, this shows that is isomorphic to the quotient of by This is the idea behind the presentation of groups by generators and relations : every finitely generated group is constructible as a quotient of a free group by some relations.

**Reference:** D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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