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.