# Regular Representations in Group are Permutations

## Theorem

Let $\struct {G, \circ}$ 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 Representation of Invertible Element is Permutation applies.

$\blacksquare$