Free group

Remarks

The identity of $F_S$ is the empty word, and the inverse of $t_1 \cdots t_n$ is $t_n^{-1} \cdots t_1^{-1}$. The set $S$ is a basis of $F_S$. Also $F_S \cong F_T$ if and only if $S$ and $T$ have the same cardinality, which is the rank. The free group of rank $1$ is the infinite cyclic group, and for $|S| \geq 2$ the centre of $F_S$ is trivial. A nontrivial finite group is never free, because each element of a basis has infinite order.