##  [Free Group](/free-group) 

  ##  [Free Group](https://algebra.quantumdictionary.io/free-group-0) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A group generated by a set S with no relations other than those forced by the group axioms: elements are reduced words in S ∪ S^{-1} and multiplication is concatenation followed by free reduction; equivalently characterized by a universal property for maps from S to groups.