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 . (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 , for some integers and .
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