Is inverse sufficient for a semigroup to be a group?

Question

If $(A,\cdot)$ is a set with a closed operation ( $a,b\in A \rightarrow a\cdot b\in A$) and associative demanding it to be inverse (from both left and right) must make it a group? as we must have identity element by the definition of inverse ($a\cdot a^{-1}=1$) so we "get the identity for free"?