Free group

In group theory, a free group is a group in which there is a generating set such that every element of the group can be writtenly uniquely as the product, or string, of generators. Every group is isomorphic to a quotient group of some free group, so understanding the properties of free groups helps us understand the structure of all groups. Free groups are also used to find the presentation of a group, a useful tool used to completely characterize the structure of a group.

To construct the free group on a set $X$ of generators, we take a second set ${\bar {X}}$ in one-to-one correspondence with X, letting the symbol $x^{-1}$ in ${\bar {X}}$ correspond to the element $x\in X$. We consider words, that is, finite sequences, over the "alphabet" $X\cup {\bar {X}}$ and take the binary operation of concatenation (juxtaposition) of words. The identity element for this operation is the empty string. (So far we have described the free monoid on the alphabet.) We define the inverse of a word to be the word obtained by reversing the order of the symbols and replacing each symbol from ${\bar {X}}$ by the corresponding symbol from $x^{-1}$ and vice versa. We finally consider words to be equivalent if they can be obtained by insertion or deletion of consecutive terms of the form $xx^{-1}$ or $x^{-1}x$. The equivalence classes now form a group, and this is the free group on X.

We define a group to be free if it is isomorphic to the free group on some generating set.

The rank of a free group is the cardinality of a generating set. A subgroup of a free group is again free, but the rank may increase. If F is a free group of finite rank n and H is a subgroup of finite index j, then the rank of H is $j(n-1)+1$.

Examples

• The free group on the empty set is the trivial group of one element.
• The free group on a single element g is the additive group of integers with $n\leftrightarrow g^{n}$.
• The free group on two generators g, h is the set of all words of the form $g^{m_{1}}h^{n_{1}}g^{m_{2}}h^{m_{2}}\cdots g^{m_{r}}h^{n_{r}}$, where all the exponents are non-zero integers except possibly $m_{1}$ and $n_{r}$.