Free Group

Let $S$ be a set . A free group on $S$ is a group $F(S)$ together with an injection $i:S\to F(S)$ such that the following universal property holds:

For every group $G$ and every function $f:S\to G$, there exists a unique group homomorphism $\widetilde f:F(S)\to G$ with $\widetilde f\circ i = f$.

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.