Free group: Difference between revisions
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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. | 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. | ||
Revision as of 16:48, 13 November 2008
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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 of generators, we take a second set in one-to-one correspondence with X, letting the symbol in correspond to the element . We consider words, that is, finite sequences, over the "alphabet" 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 by the corresponding symbol from 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 or . The equivalence classes now form a group, and this is the free group on X.
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 .