The infinite cyclic group is a free group of rank , freely generated by either or .
An example of a free group of rank is the multiplicative group of integer matrices generated by
If is a free group freely generated by a set , where , then for distinct the set generates a free group of countably infinite rank.
If a free group is freely generated by , then is a minimal generating set for , and no set of smaller cardinality than can generate . It follows that if is freely generated by both and , then . So the rank of a free group is a well-defined concept, and free groups of different ranks are non-isomorphic.
Every group is a homomorphic image of some free group. More precisely, if is a group generated by a set of cardinality , then is a homomorphic image of every free group of rank or more.
For any set , the following construction gives a free group of rank .
Let be a set with elements for in some index set . We refer to as an alphabet and the elements of as letters. A syllable is a symbol of the form for . It is customary to write for . Define a word to be a finite sequence of syllables. For example,
is a five-syllable word. Notice that there exists a unique empty word, i.e., the word with no syllables, usually written simply as . Denote the set of all words formed from elements of by .
Define a binary operation, called the product, on by concatenation of words. To illustrate, if and are elements of then their product is simply . This gives the structure of a monoid. The empty word acts as a right and left identity (http://planetmath.org/LeftIdentityAndRightIdentity) in , and is the only element which has an inverse. In order to give the structure of a group, two more ideas are needed.
If is a word where are also words and is some element of , an elementary contraction of type I replaces the occurrence of by . Thus, after this type of contraction we get another word . If is a word, an elementary contraction of type II replaces the occurrence of by which results in . In either of these cases, we also say that is obtained from by an elementary contraction, or that is obtained from by an elementary expansion.
Call two words equivalent (denoted ) if one can be obtained from the other by a finite sequence of elementary contractions or expansions. This is an equivalence relation on . Let be the set of equivalence classes of words in . Then is group under the operation
where . The inverse of an element is obtained by reversing the order of the syllables of and changing the sign of each syllable. For example, if , then .
It can be shown that is a free group freely generated by the set . Moreover, a group is free if and only if it is isomorphic to for some set .
|Date of creation||2013-03-22 12:28:39|
|Last modified on||2013-03-22 12:28:39|
|Last modified by||yark (2760)|