Regular Representations in Semigroup are Permutations then Structure is Group

Theorem
Let $\struct {S, \circ}$ be a semigroup.

For $a \in S$, let $\lambda_a: S \to S$ and $\rho_a: S \to S$ denote the left regular representation and right regular representation with respect to $a$ respectively:

For all $a$ in $S$, let $\lambda_a$ be a permutation on $S$.

Let there exist $b$ in $S$ such that $\rho_b$ is a permutation on $S$.

Then $\struct {S, \circ}$ is a group.