The free group on a given set S consists of all words that can be built from members of S, where two words count as different unless their equality follows purely from the group axioms. The elements of S are called generators, and the number of generators is the rank of the free group. A group is called free if it is isomorphic to the free group on some subset of its own elements, meaning every element of the group can be written in exactly one way as a product of finitely many generators and their inverses. Free groups are defined by a universal property and are closely related to, but distinct from, free abelian groups.
Facts
Classification
Object KindStructure or Algebraic Object 1 Connections
In Branch
Source Wolfram MathWorld: Free Group
Is Kind Of Object
Sources
1. Wikipedia: Free group
History section
The algebraic study of free groups was initiated by Jakob Nielsen in 1924, who gave them their name and established many of their basic properties.
Introduction section
the group of all reduced words in S, with concatenation of words (followed by reduction if necessary) as group operation.
View the SourceWolfram MathWorld: Free Group
Wolfram MathWorldIn Branch: Group Theory, Subject classificationsQuote, In Branch: Group Theory, Subject classifications
Algebra Group Theory
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.