Free Group - Facts and Theorems

Facts and Theorems

Some properties of free groups follow readily from the definition:

1. Any group G is the homomorphic image of some free group F(S). Let S be a set of generators of G. The natural map f: F(S) → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F(S). The kernel of f is a set of relations in the presentation of G. If S can be chosen to be finite here, then G is called finitely generated.
2. If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consists only of the identity element).
3. Two free groups F(S) and F(T) are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k.
4. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1.

A few other related results are:

1. The Nielsen–Schreier theorem: Any subgroup of a free group is free.
2. A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks.
3. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators for non-zero m and n.
4. The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of all countable ranks.
5. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank (given by 1 plus the Euler characteristic of the quotient graph).
6. The Cayley graph of a free group of finite rank, with respect to a free generating set, is a tree on which the group acts freely, preserving the orientation.
7. The groupoid approach to these results, given in the work by P.J. Higgins below, is kind of extracted from an approach using covering spaces. It allows more powerful results, for example on Grushko's theorem, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph.
8. Grushko's theorem has the consequence that if a subset B of a free group F on n elements generates F and has n elements, then B generates F freely.