Free Groups, Free Abelian Groups

Free Abelian Groups

Let's build a free abelian group, which is a free object in the category of abelian groups.

Let S be a set of letters, and build words from these letters and their inverses, but this time like letters and their inverses can be brought together. This is because symbols commute past each other, as you would expect from an abelian group. For notational convenience, a letter is paired with an integer, representing the number of copies of that letter. The integer is negative if the inverse letter is used. The integer may appear as an exponent, or a coefficient, as a matter of taste. Remember that additive notation is often used when dealing with abelian groups, as shown by the second example below.

x3y5 * xy4 = x4y9

3x+5y + x+4y = 4x+9y

The free abelian group on S is equal to the direct sum of copies of Z - one copy for each generator in S. The exponents, or coefficients, are the integer entries drawn from each copy of Z.

If S is mapped into any abelian group G, F follows S, and wraps around G uniquely. Thus the free abelian group is indeed a free object.

Every abelian group is a Z module, and Z is an integral domain and a pid. Theorems about modules over an integral domain, or a pid, are in force. In particular, the rank of F is well defined. There is one free abelian group, up to isomorphism, for each cardinal number of generators, and groups of different rank are not isomorphic. Furthermore, a finitely generated abelian group is a direct product of cyclic groups, Z or Zn.

Commutator Subgroup

Let F be the free group on S, and include a relation xy = yx for every pair of generators x and y in S. Since x and y commute past each other, the quotient group is abelian. The kernel, spanned by these commutator relations, is called the commutator subgroup of F. Express the relations as relators, and you will see that the kernel is already normal. Conjugate xy/x/y by v and get vxy/y/x/v, which is the commutator of vx and y.

If F′ is F mod its commutator subgroup, F′ is free abelian. Conversely, a free abelian group F′ on S is the image of a free group F on S, and the kernel includes, and is spanned by, the commutators xy = yx. The generators of F and the generators of F′ correspond.

Invariant Dimension Property

As shown above, the rank of a free abelian group is well defined. Use this to make the same assertion about free groups.

If two free groups are isomorphic, yet they have different ranks, i.e. different numbers of generators, mod out by their commutator subgroups and find two isomorphic free abelian groups with different ranks. This is a contradiction, hence the rank of a free group is well defined. The free group on 3 letters is not equivalent to the free group on 2 letters, or the free group on an infinite set of letters.

Mapping Onto F

If a homomorphism takes the free group E onto the free group F, E cannot have lesser rank. A commutator xy/x/y in E becomes a commutator in F. Treat the commutator subgroups as kernels. Since the kernel of E maps into the kernel of F, the map on quotient groups is well defined. The abelianization of E maps onto the abelianization of F, and these are free Z modules. The dimension of the former is at least the dimension of the latter; hence the rank of E is at least the rank of F.

We'll see later on that F can fit inside a free group of lesser rank. This is quite different from the abelian case.