The *free group* on k generators is the group **F**_{k} = ⟨a_{1},...,a_{k}⟩. See generators and relations for groups for precise definition and explanation of what "generated" means here (or see the short explanation below). There are no relations for this group. One can also define free groups on infinite sets of generators in the obvious manner; however, note that products of these generators are still finite (we have no topology, hence no convergence).

Elements of the group have a canonical form a_{i1}^{n1}...a_{im}^{nm}, where 1≤i_{j}≤k is the index of some generator, 0≠n_{j}∈**Z** is a nonzero integer, and i_{j}≠i_{j+1}.
Any element can be translated to canonical form by combining adjacently equal a_{i}'s. For instance,

a_{2} a_{1} a_{3}^{2} a_{3}^{-4} a_{1}^{2} =
a_{2} a_{1} a_{3}^{-2} a_{1}^{2}.

Thus, the

word problem on the free group is (easily)

solvable by

computer program.

## Making groups from the free group

Any group with k generators is a quotient group of **F**_{k}. The relations of the group *are* a normal subgroup N, and the group is simply **F**_{k}/N.

Obviously, **F**_{j} is a subgroup (as well as a quotient
group) of **F**_{k} whenever j≤k. Somewhat amazingly, the
reverse also holds! We shall show that **F**_{k} is a subgroup
of **F**_{2} -- thus, study of this one group encompasses all
of (finite, and even countable) group theory!

To generate a copy **F**_{k}≅G≤**F**_{2}=⟨a,b⟩, we look
at the subgroup generated by the elements

⟨e_{1}=a^{1}ba^{-1},...,e_{k}=a^{k}ba^{-k}⟩.

Then we have that e

_{i}^{m}=a

^{i}b

^{m}a

^{-i},
so we can readily identify powers of our generators. And

multiplication
behaves nicely: if i≠j, then

e_{i}^{m}e_{j}^{n} = a^{i}b^{m}a^{j-i}b^{n}a^{-j}.

Thus, we can easily "

decode" any finite

product of e

_{i}'s when phrased as a product of a's and b's. The rigourous argument to show
our G is free proceeds in this fashion.

The free group can also be found as a quotient in numerous places. Perhaps the
most famous is the identification of a subgroup of PSL_{2}(**Z**) (itself a quotient group of the group SL_{2}(**Z**) of integer 2×2 matrices with determinant 1) as the free group **F**_{2}. This identification is what makes hyperbolic geometry interesting.