Editing Free group (section)

==Facts and theorems==
Some properties of free groups follow readily from the definition:

#Any group ''G'' is the homomorphic image of some free group ''F<sub>S</sub>''. Let ''S'' be a set of ''[[Generating set of a group|generators]]'' of ''G''. The natural map ''φ'': ''F<sub>S</sub>'' → ''G'' is an [[epimorphism]], which proves the claim. Equivalently, ''G'' is isomorphic to a [[quotient group]] of some free group ''F<sub>S</sub>''. If ''S'' can be chosen to be finite here, then ''G'' is called [[finitely generated group|finitely generated]]. The kernel Ker(''φ)'' is the set of all ''relations'' in the [[Presentation of a group|presentation]] of ''G''; if Ker(''φ)'' can be generated by the conjugates of finitely many elements of ''F'', then ''G'' is finitely presented.
#If ''S'' has more than one element, then ''F<sub>S</sub>'' is not [[abelian group|abelian]], and in fact the [[center of a group|center]] of ''F<sub>S</sub>'' is trivial (that is, consists only of the identity element).
#Two free groups ''F<sub>S</sub>'' and ''F<sub>T</sub>'' 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''.
#A free group of finite rank ''n'' > 1 has an [[exponential growth|exponential]] [[growth rate (group theory)|growth rate]] of order 2''n'' − 1.

A few other related results are:
#The [[Nielsen–Schreier theorem]]:  Every [[subgroup]] of a free group is free. Furthermore, if the free group ''F'' has rank ''n'' and the subgroup ''H'' has [[Index of a subgroup|index]] ''e'' in ''F'', then ''H'' is free of rank 1 + ''e''(''n–''1).
#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 set|countable]] ranks.
#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 [[commutator]]s [''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>] for non-zero ''m'' and ''n''.
#The free group in two elements is [[SQ universal]]; the above follows as any SQ universal group has subgroups of all countable ranks.
#Any group that [[Group action (mathematics)|acts]] on a tree, [[free action|freely]] and preserving the [[oriented graph|orientation]], is a free group of countable rank (given by 1 plus the [[Euler characteristic]] of the [[Group action (mathematics)|quotient]] [[graph theory|graph]]).
#The [[Cayley graph]] of a free group of finite rank, with respect to a free generating set, is a [[tree (graph theory)|tree]] on which the group acts freely, preserving the orientation. As a topological space (a one-dimensional [[simplicial complex]]), this Cayley graph Γ(''F'') is [[Contractible space|contractible]]. For a finitely presented group ''G,'' the natural homomorphism defined above, ''φ'' : ''F'' → ''G'', defines a [[Covering space|covering map]] of Cayley graphs ''φ*'' : Γ(''F'') → Γ(''G''), in fact a universal covering. Hence, the [[fundamental group]] of the Cayley graph Γ(''G'') is isomorphic to the kernel of ''φ'', the normal subgroup of relations among the generators of ''G''. The extreme case is when ''G'' = {''e''}, the trivial group, considered with as many generators as ''F'', all of them trivial; the Cayley graph Γ(''G'') is a bouquet of circles, and its fundamental group is ''F'' itself.
#Any subgroup of a free group, <math>H \subset F</math>, corresponds to a covering space of the bouquet of circles, namely to the [[Schreier coset graph]] of ''F''/''H''. This can be used to give a topological proof of the Nielsen-Schreier theorem above.
#The [[groupoid]] approach to these results, given in the work by P.J. Higgins below, is related to the use of [[covering space]]s above. 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.
# [[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.