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.

$\blacksquare$

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$: Example $18$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 28 \beta$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 35.8$: Elementary consequences of the group axioms