Regular Representations in Group are Permutations

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $a \in G$ be any element of $G$.

Then the left regular representation $\lambda_a$ and the right regular representation $\rho_a$ are permutations of $G$.

Proof
This follows directly from the fact that all elements of a group are by definition invertible.

Therefore the result Regular Representations of Invertible Elements are Permutations applies.