What is "free group"

n. (context mathematics English) A group (set with a particular kind of binary operation) that has a presentation without relators; equivalently, a free product of some number of copy of ℤ.

In mathematics, the free group F over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st = suut, but s ≠ t for s,t,u∈S). The members of S are called generators of F. An arbitrary group G is called free if it is isomorphic to F for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suut).

A related but different notion is a free abelian group, both notions are particular instances of a free object from universal algebra.