Free Group

Definition

Let $S$ be a set of symbols. The free group on $S$, written $F(S)$, is the group whose elements are all finite words made from symbols in $S$ and their formal inverses, with only the cancellation rules

$$s{s}^{-1}=e,s^{-1}s=e$$

for every $s\in S$.

NOTE

“Free” means there are no relations between the generators except the group axioms. So unless a word can be simplified by cancelling adjacent inverse pairs, it represents a different group element.

Example

In the free group $F(a,b)$, we may build words like

$$a,b,a^{-1},ab,ab{a}^{-1},a{b}^{-1}{a}^{-1}b.$$ $$(a{b}^{-1})(b{a}^{-1})=a{b}^{-1}b{a}^{-1}=a{a}^{-1}=e.$$

Warning

$F(a,b)$ is not abelian.

Nielsen–Schreier theorem

Nielsen–Schreier theorem

every subgroup of a free group is free