Free group

A free group is a type of group that is of particular importance in combinatorics. Let $I$ be any nonempty index set. Informally, a free group on $I$ is the collection of finite strings of characters from the collection $\{X_i,X_i^{-1}:i\in I\}$ subject only to the criterion that $X_iX_i^{-1}=X_i^{-1}X_i=1$ (the group identity). The group operation is concatenation. An example of an element of the free group on two letters is $X_1X_2^{-1}X_1^{-1}X_2^3$ (where by $X_2^3$ we mean $X_2X_2X_2$ ).

More formally, free groups are defined by universal properties. A group $F$ is called a free group on $I$ if there is a function $\phi:I\to F$ so that for any group $G$ and a function $\theta:I\to G$ , there is a unique group homomorphism $\psi:F\to G$ so that $\theta=\psi\phi$ , i.e. so that $\theta(i)=\psi\circ\phi(i)$ for all $i\in I$ . 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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Free group