Free Group - Knowlpedia

Let $S$ be a set. A free group on $S$ is a group $F(S)$ with a function $i:S\to F(S)$ such that, for every group $G$ and function $f:S\to G$ , there is a unique group homomorphism $\widetilde f:F(S)\to G$ satisfying $\widetilde f\circ i=f$ .

The universal property implies that $i$ is injective and that $i(S)$ generates $F(S)$ .

Remarks

Concretely, elements of $F(S)$ can be represented by reduced words in symbols from $S$ and their formal inverses. Free groups are the starting point for group presentations: adding relations corresponds to taking a quotient.

Examples

The free group on one generator is isomorphic to $\mathbb{Z}$ (send the generator to $1$ ).
The free group on two generators has elements represented by reduced words in $a^{\pm1},b^{\pm1}$ (e.g. $ab^{-1}a^{-1}b$ ).
If $S=\varnothing$ , then $F(S)$ is the trivial group.