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
(the group identity). The group operation is concatenation. An example of an element of the free group on two letters is
(where by
we mean
).
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.
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