everybody. I got this exercise from Jacobson.
Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group.
Proof: every element of $M$ has the form $s_1,s_2,...,s_n$ which is invertible with inverse $s^{-1}_1,s^{-1}_2,...,s^{-1}_n \in M$. Hence, $M$ is a group.