Free group
A free group is a type of group that is of particular importance in combinatorics. Let 
be any nonempty index set. Informally, a free group on is the collection of finite strings of characters from the collection 
subject only to the criterion that 
where 
is the group identity and is equal to the empty string. The group operation is concatenation.
An example of an element of the free group on 
is 
(where by 
we mean 
).
The inverse of a given element of a free group can be found by reversing the string and the sign of all exponents. Thus, the string given above has inverse 
. This is easy to check: we just apply the group operation to the two strings to get $\displaystyle X_2^{-3}X_1X_2X_1^{-1}X_1X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2\cdot1\cdot X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1\cdot 1\cdot X_1^{-1}X_2^3 = X_2^{-3}X_1X_1^{-1}X_2^3 = X_2^{-3}\cdot 1\cdotX_2^3 = X_2^{-3}X_2^3 = 1$ (Error compiling LaTeX. Unknown error_msg). (Note that we implicitly used the associativity of the group operation repeatedly in this process.) Showing that this string is a right inverse is equally straightforward. The proof that this holds for every string is by induction using the same idea.
More formally, free groups are defined by universal properties. A group 
is called a free group on if there is a function 
so that for any group 
and a function 
, there is a unique group homomorphism 
so that 
, i.e. so that 
for all 
. We often like to draw a diagram to represent this relationship; however WE DON'T HAVE THE xy PACKAGE INCLUDED, SO I CAN'T TeX IT.
Commutivity in Free Groups
The free group over has "as little structure as possible." Thus, we can show that two elements of the free group commute only when it's "necessary":
If 
are elements of the free group over a set and 
, then 
or 
.
Proof
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